![Page 1: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/1.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Exponentially-fitted methods applied tofourth-order boundary value problems
M. Van Daele, D. Hollevoet and G. Vanden Berghe
Department of Applied Mathematics and Computer Science
Sixth International Conference of Numerical Analysis andApplied Mathematics, Kos, 2008
![Page 2: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/2.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Outline
IntroductionAn exampleExponential fitting
Fourth-order boundary value problemsThe problemExponentially-fitted methodsParameter selection
Numerical examplesFirst exampleSecond example
Conclusions
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
Introduction
In the past 15 years, our research group has constructedmodified versions of well-known
• linear multistep methods
• Runge-Kutta methods
Aim : build methods which perform very good when the solutionhas a known exponential of trigonometric behaviour.
![Page 4: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/4.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Linear multistep methods
A well known method to solve
y ′′ = f (y) y(a) = ya y ′(a) = y ′a
is the Numerov method (order 4)
yn+1 − 2 yn + yn−1 =1
12h2 (f (yn−1) + 10 f (yn) + f (yn+1))
Construction :impose L[z(t); h] = 0 for z(t) ∈ S = {1, t , t2, t3, t4} where
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
![Page 5: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/5.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Linear multistep methods
A well known method to solve
y ′′ = f (y) y(a) = ya y ′(a) = y ′a
is the Numerov method (order 4)
yn+1 − 2 yn + yn−1 =1
12h2 (f (yn−1) + 10 f (yn) + f (yn+1))
Construction :impose L[z(t); h] = 0 for z(t) ∈ S = {1, t , t2, t3, t4} where
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
![Page 6: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/6.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
A model problem
Consider the initial value problem
y ′′ + ω2 y = g(y) y(a) = ya y(a) = y ′a .
If |g(y)| ≪ |ω2 y | then
y(t) ≈ α cos(ω t + φ)
To mimic this oscillatory behaviour, one could replacepolynomials by trigonometric (in the complex case :exponential) functions.
![Page 7: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/7.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
A model problem
Consider the initial value problem
y ′′ + ω2 y = g(y) y(a) = ya y(a) = y ′a .
If |g(y)| ≪ |ω2 y | then
y(t) ≈ α cos(ω t + φ)
To mimic this oscillatory behaviour, one could replacepolynomials by trigonometric (in the complex case :exponential) functions.
![Page 8: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/8.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
A model problem
Consider the initial value problem
y ′′ + ω2 y = g(y) y(a) = ya y(a) = y ′a .
If |g(y)| ≪ |ω2 y | then
y(t) ≈ α cos(ω t + φ)
To mimic this oscillatory behaviour, one could replacepolynomials by trigonometric (in the complex case :exponential) functions.
![Page 9: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/9.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Numerov methodConstruction : impose L[z(t); h] = 0 for z(t) ∈ S with
S = {1, t , t2, sin(ω t), cos(ω t)}
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
yn+1 − 2 yn + yn−1 = h2 (λf (yn−1) + (1 − 2 λ) f (yn) + λ f (yn+1))
λ =1
4 sin2 θ2
−1θ2 θ := ω h
=1
12+
1240
θ2 +1
6048θ4 + . . .
![Page 10: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/10.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Numerov methodConstruction : impose L[z(t); h] = 0 for z(t) ∈ S with
S = {1, t , t2, sin(ω t), cos(ω t)}
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
yn+1 − 2 yn + yn−1 = h2 (λf (yn−1) + (1 − 2 λ) f (yn) + λ f (yn+1))
λ =1
4 sin2 θ2
−1θ2 θ := ω h
=1
12+
1240
θ2 +1
6048θ4 + . . .
![Page 11: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/11.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Numerov methodConstruction : impose L[z(t); h] = 0 for z(t) ∈ S with
S = {1, t , t2, sin(ω t), cos(ω t)}
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
yn+1 − 2 yn + yn−1 = h2 (λf (yn−1) + (1 − 2 λ) f (yn) + λ f (yn+1))
λ =1
4 sin2 θ2
−1θ2 θ := ω h
=1
12+
1240
θ2 +1
6048θ4 + . . .
![Page 12: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/12.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Numerov methodConstruction : impose L[z(t); h] = 0 for z(t) ∈ S with
S = {1, t , t2, sin(ω t), cos(ω t)}
L[z(t); h] := z(t + h) + α0 z(t) + α−1 z(t − h)
−h2 (
β1 z ′′(t + h) + β0 z ′′(t) + β−1 z ′′(t − h))
yn+1 − 2 yn + yn−1 = h2 (λf (yn−1) + (1 − 2 λ) f (yn) + λ f (yn+1))
λ =1
4 sin2 θ2
−1θ2 θ := ω h
=1
12+
1240
θ2 +1
6048θ4 + . . .
![Page 13: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/13.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 14: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/14.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 15: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/15.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 16: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/16.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 17: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/17.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 18: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/18.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF methodsGeneralisation : to determine the coefficients of a method, weimpose conditions on a linear functional. These conditions arerelated to the fitting space S which contains
• polynomials :{tq|q = 0, . . . , K}
• exponential or trigonometric functions, multiplied withpowers of t :
{tq exp(±µt)|q = 0, . . . , P}
or, with ω = i µ,
{tq cos(ωt), tq sin(ωt)|q = 0, . . . , P}
EF method can be characterized by the couple (K , P)Classical method : P = −1number of basis functions : M= 2 P + K + 3
![Page 19: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/19.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Examples
M = 2 P + K + 3
(K , P)
M = 2 M = 4 M = 6 M = 8 M = 10(1,−1) (3,−1) (5,−1) (7,−1) (9,−1)(−1, 1) (1, 0) (3, 0) (5, 0) (7, 0)
(−1, 1) (1, 1) (3, 1) (5, 1)(−1, 2) (1, 2) (3, 2)
(−1, 3) (1, 3)(−1, 4)
(1, 2) =⇒ S ={
1, t , exp(±µt), t exp(±µt), t2 exp(±µt)}
![Page 20: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/20.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Examples
M = 2 P + K + 3
(K , P)
M = 2 M = 4 M = 6 M = 8 M = 10(1,−1) (3,−1) (5,−1) (7,−1) (9,−1)(−1, 1) (1, 0) (3, 0) (5, 0) (7, 0)
(−1, 1) (1, 1) (3, 1) (5, 1)(−1, 2) (1, 2) (3, 2)
(−1, 3) (1, 3)(−1, 4)
(1, 2) =⇒ S ={
1, t , exp(±µt), t exp(±µt), t2 exp(±µt)}
![Page 21: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/21.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Exponential FittingL. Ixaru and G. Vanden BergheExponential fittingKluwer Academic Publishers, Dordrecht, 2004
η−1(Z ) =
{
cos(|Z |1/2) if Z < 0cosh(Z 1/2) if Z ≥ 0
η0(Z ) =
sin(|Z |1/2)/|Z |1/2 if Z < 01 if Z = 0sinh(Z 1/2)/Z 1/2 if Z > 0
Z := (µ h)2 = −(ω h)2
ηn(Z ) :=1Z
[ηn−2(Z ) − (2n − 1)ηn−1(Z )], n = 1, 2, 3, . . .
η′n(Z ) =12ηn+1(Z ), n = 1, 2, 3, . . .
![Page 22: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/22.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Exponential FittingL. Ixaru and G. Vanden BergheExponential fittingKluwer Academic Publishers, Dordrecht, 2004
η−1(Z ) =
{
cos(|Z |1/2) if Z < 0cosh(Z 1/2) if Z ≥ 0
η0(Z ) =
sin(|Z |1/2)/|Z |1/2 if Z < 01 if Z = 0sinh(Z 1/2)/Z 1/2 if Z > 0
Z := (µ h)2 = −(ω h)2
ηn(Z ) :=1Z
[ηn−2(Z ) − (2n − 1)ηn−1(Z )], n = 1, 2, 3, . . .
η′n(Z ) =12ηn+1(Z ), n = 1, 2, 3, . . .
![Page 23: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/23.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Exponential FittingL. Ixaru and G. Vanden BergheExponential fittingKluwer Academic Publishers, Dordrecht, 2004
η−1(Z ) =
{
cos(|Z |1/2) if Z < 0cosh(Z 1/2) if Z ≥ 0
η0(Z ) =
sin(|Z |1/2)/|Z |1/2 if Z < 01 if Z = 0sinh(Z 1/2)/Z 1/2 if Z > 0
Z := (µ h)2 = −(ω h)2
ηn(Z ) :=1Z
[ηn−2(Z ) − (2n − 1)ηn−1(Z )], n = 1, 2, 3, . . .
η′n(Z ) =12ηn+1(Z ), n = 1, 2, 3, . . .
![Page 24: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/24.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Exponential FittingL. Ixaru and G. Vanden BergheExponential fittingKluwer Academic Publishers, Dordrecht, 2004
η−1(Z ) =
{
cos(|Z |1/2) if Z < 0cosh(Z 1/2) if Z ≥ 0
η0(Z ) =
sin(|Z |1/2)/|Z |1/2 if Z < 01 if Z = 0sinh(Z 1/2)/Z 1/2 if Z > 0
Z := (µ h)2 = −(ω h)2
ηn(Z ) :=1Z
[ηn−2(Z ) − (2n − 1)ηn−1(Z )], n = 1, 2, 3, . . .
η′n(Z ) =12ηn+1(Z ), n = 1, 2, 3, . . .
![Page 25: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/25.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Choice of ω
• local optimizationbased on local truncation error (lte)ω is step-dependent
• global optimizationPreservation of geometric properties (periodicity, energy,. . . )ω is constant over the interval of integration
![Page 26: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/26.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Choice of ω
• local optimizationbased on local truncation error (lte)ω is step-dependent
• global optimizationPreservation of geometric properties (periodicity, energy,. . . )ω is constant over the interval of integration
![Page 27: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/27.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Fourth-order boundary value problem
y (4) = F (t , y) a ≤ t ≤ b
y(a) = A1 y ′′(a) = A2
y(b) = B1 y ′′(b) = B2
• special case : y (4) + f (t) y = g(t)
• mathematical modeling of viscoelastic and inelastic flows,deformation of beams, plate deflection theory, . . .
• work by Doedel, Usmani, Agarwal, Cherruault et al., VanDaele et al., . . .
• finite differences, B-splines, . . .
![Page 28: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/28.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Fourth-order boundary value problem
y (4) = F (t , y) a ≤ t ≤ b
y(a) = A1 y ′′(a) = A2
y(b) = B1 y ′′(b) = B2
• special case : y (4) + f (t) y = g(t)
• mathematical modeling of viscoelastic and inelastic flows,deformation of beams, plate deflection theory, . . .
• work by Doedel, Usmani, Agarwal, Cherruault et al., VanDaele et al., . . .
• finite differences, B-splines, . . .
![Page 29: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/29.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Fourth-order boundary value problem
y (4) = F (t , y) a ≤ t ≤ b
y(a) = A1 y ′′(a) = A2
y(b) = B1 y ′′(b) = B2
• special case : y (4) + f (t) y = g(t)
• mathematical modeling of viscoelastic and inelastic flows,deformation of beams, plate deflection theory, . . .
• work by Doedel, Usmani, Agarwal, Cherruault et al., VanDaele et al., . . .
• finite differences, B-splines, . . .
![Page 30: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/30.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Fourth-order boundary value problem
y (4) = F (t , y) a ≤ t ≤ b
y(a) = A1 y ′′(a) = A2
y(b) = B1 y ′′(b) = B2
• special case : y (4) + f (t) y = g(t)
• mathematical modeling of viscoelastic and inelastic flows,deformation of beams, plate deflection theory, . . .
• work by Doedel, Usmani, Agarwal, Cherruault et al., VanDaele et al., . . .
• finite differences, B-splines, . . .
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
Fourth-order boundary value problem
y (4) = F (t , y) a ≤ t ≤ b
y(a) = A1 y ′′(a) = A2
y(b) = B1 y ′′(b) = B2
• special case : y (4) + f (t) y = g(t)
• mathematical modeling of viscoelastic and inelastic flows,deformation of beams, plate deflection theory, . . .
• work by Doedel, Usmani, Agarwal, Cherruault et al., VanDaele et al., . . .
• finite differences, B-splines, . . .
![Page 32: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/32.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
The formulae
tj = a + j h, j = 0, 1, . . . N + 1 N ≥ 3 h :=b − aN + 1
• central formula for j = 2, . . . , N − 1
yj−2 + a1 yj−1 + a0 yj + a1 yj+1 + yj+2 =
h4 (
b2 Fj−2 + b1 Fj−1 + b0 Fj + b1 Fj+1 + b2 Fj+2)
whereby yj is approximate value of y(tj) and Fj := F (tj , yj).
• begin formula
y0 + α1 y1 + α2 y2 + a3 y3 =
γ h2 y ′′0 + h4 (β0 F0 + β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5)
• end formula
![Page 33: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/33.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
The formulae
tj = a + j h, j = 0, 1, . . . N + 1 N ≥ 3 h :=b − aN + 1
• central formula for j = 2, . . . , N − 1
yj−2 + a1 yj−1 + a0 yj + a1 yj+1 + yj+2 =
h4 (
b2 Fj−2 + b1 Fj−1 + b0 Fj + b1 Fj+1 + b2 Fj+2)
whereby yj is approximate value of y(tj) and Fj := F (tj , yj).
• begin formula
y0 + α1 y1 + α2 y2 + a3 y3 =
γ h2 y ′′0 + h4 (β0 F0 + β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5)
• end formula
![Page 34: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/34.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
The formulae
tj = a + j h, j = 0, 1, . . . N + 1 N ≥ 3 h :=b − aN + 1
• central formula for j = 2, . . . , N − 1
yj−2 + a1 yj−1 + a0 yj + a1 yj+1 + yj+2 =
h4 (
b2 Fj−2 + b1 Fj−1 + b0 Fj + b1 Fj+1 + b2 Fj+2)
whereby yj is approximate value of y(tj) and Fj := F (tj , yj).
• begin formula
y0 + α1 y1 + α2 y2 + a3 y3 =
γ h2 y ′′0 + h4 (β0 F0 + β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5)
• end formula
![Page 35: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/35.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
The formulae
tj = a + j h, j = 0, 1, . . . N + 1 N ≥ 3 h :=b − aN + 1
• central formula for j = 2, . . . , N − 1
yj−2 + a1 yj−1 + a0 yj + a1 yj+1 + yj+2 =
h4 (
b2 Fj−2 + b1 Fj−1 + b0 Fj + b1 Fj+1 + b2 Fj+2)
whereby yj is approximate value of y(tj) and Fj := F (tj , yj).
• begin formula
y0 + α1 y1 + α2 y2 + a3 y3 =
γ h2 y ′′0 + h4 (β0 F0 + β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5)
• end formula
![Page 36: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/36.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = −1 : L[y ] = 0 for y ∈ S ={
1, t , t2, . . . , tM−1}
![Page 37: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/37.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = −1 : L[y ] = 0 for y ∈ S ={
1, t , t2, . . . , tM−1}
M = 10 :
yp−2 − 4 yp−1 + 6 yp − 4 yp+1 + yp+2 =
h4
720
(
−y (4)p−2 + 124 y (4)
p−1 + 474 y (4)p + 124 y (4)
p+1 − y (4)p+2
)
L[y ](t) =1
3024h10 y (10)(t) + O(h12)
![Page 38: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/38.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = −1 : L[y ] = 0 for y ∈ S ={
1, t , t2, . . . , tM−1}
M = 8 and b2 = 0 :
yp−2 −4 yp−1 +6 yp −4 yp+1 +yp+2 =h4
6
(
y (4)p−1 + 4 y (4)
p + y (4)p+1
)
L[y ](t) = −1
720h8 y (8)(t) + O(h10)
![Page 39: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/39.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = −1 : L[y ] = 0 for y ∈ S ={
1, t , t2, . . . , tM−1}
M = 6 and b1 = b2 = 0 :
yp−2 − 4 yp−1 + 6 yp − 4 yp+1 + yp+2 = h4 y (4)p
L[y ](t) =16
h6 y (6)(t) + O(h8)
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 0 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), 1, t , t2, . . . , tM−3}
![Page 41: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/41.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 0 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), 1, t , t2, . . . , tM−3}
M = 10 :
yp−2 − 4 yp−1 + 6 yp − 4 yp+1 + yp+2 =
h4(
b2 y (4)p−2 + b1 y (4)
p−1 + b0 y (4)p + b1 y (4)
p+1 + b2 y (4)p+2
)
b0 =4 cos2 θ − 2 − 11 cos θ
6 (cos θ − 1)2+
6
θ4b1 =
cos2 θ + 5
6 (cos θ − 1)2−
4
θ4b2 = −
cos θ + 2
12 (cos θ − 1)2+
1
θ4
L[y ](t) =1
3024h10
(
y (10)(t) + ω2 y (8)(t))
+ O(h12)
![Page 42: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/42.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 0 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), 1, t , t2, . . . , tM−3}
M = 8 and b2 = 0 :
yp−2−4 yp−1+6 yp−4 yp+1+yp+2 = h4(
b1 y (4)p−1 + b0 y (4)
p + b1 y (4)p+1
)
b0 =cos θ
cos θ − 1+
4 (1 − cos θ)
θ4b1 =
1
2 (1 − cos θ)+
2 (cos θ − 1)
θ4
L[y ](t) = −1
720h8
(
y (8)(t) + ω2 y (6)(t))
+ O(h10)
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 0 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), 1, t , t2, . . . , tM−3}
M = 6 and b1 = b2 = 0 :
yp−2 − 4 yp−1 + 6 yp − 4 yp+1 + yp+2 =sin4(θ/2)
(θ/2)4 h4 y (4)p
L[y ](t) =16
h6 (y (6)(t) + ω2 y (4)(t)) + O(h8)
![Page 44: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/44.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 1 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), t cos(ω t), t sin(ω t), 1, t , t2, . . . , tM−5}
![Page 45: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/45.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
EF Central formula
L[y ] := y(t − 2 h) + a1 y(t − h) + a0 y(t) + a1 y(t + h) + y(t + 2 h)
−h4(
b2 y (4)(t − 2 h) + b1 y (4)(t − h) + b0 y (4)(t) + b1 y (4)(t + h) + b2 y (4)(t + 2 h))
P = 1 : L[y ] = 0 for y ∈ S ={
cos(ω t), sin(ω t), t cos(ω t), t sin(ω t), 1, t , t2, . . . , tM−5}
M = 6 and b1 = b2 = 0 :
yp−2 + a1 yp−1 + a0 yp + a1 yp+1 + yp+2 = b0 h4 y (4)p
a0 = 2−8 sin2 θ + θ (4 cos θ − 1) sin θ − 4 cos θ + 4
θ sin θ + 4 cos θ − 4a1 = −4
sin θ (θ cos θ − 2 sin θ)
θ sin θ + 4 cos θ − 4
b0 = 4sin θ
(
sin2 θ − 2 + 2 cos θ)
θ3 (θ sin θ + 4 cos θ − 4)
L[y ](t) =16
h6 (y (6)(t) + 2 ω2 y (4)(t) + ω4 y (2)(t)) + O(h8)
![Page 46: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/46.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Coefficients of Central formula M = 6
0 2.0375 4.075−4
6
16
θ
a 0
P=−1P=0P=1P=2
0 2.0375 4.075−14
−4
6
θ
a 1
P=−1P=0P=1P=2
0 2.0375 4.0750
1
2
θ
b 0
P=−1P=0P=1P=2
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
Coefficients of Central formula M = 8
0 1 25.8
6
6.2
θ
a 0
P=−1P=0P=1P=2P=3
0 1 2−4.2
−4
−3.8
θ
a 1
P=−1P=0P=1P=2P=3
0 1 20.647
0.667
0.687
θ
b 0
P=−1P=0P=1P=2P=3
0 1 20.147
0.167
0.187
b 1
P=−1P=0P=1P=2P=3
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
Coefficients of Central formula M = 10
0 1 25.8
6
6.2
θ
a 0
P=−1P=0P=1P=2P=3P=4
0 1 2−4.2
−4
−3.8
θ
a 1
P=−1P=0P=1P=2P=3P=4
0 1 20.639
0.659
0.679
θ
b 0
0 1 20.153
0.173
0.193
b 1
0 1 2−0.021
−0.001
0.019
b 2
![Page 49: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/49.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula : coefficientsE.g. b0 in case M = 6In closed form . . .
• P = −1 :b0 = 1
• P = 0 :
b0 = 4(cos θ − 1)2
θ4
• P = 1 :
b0 = −4sin θ (cos θ − 1)2
θ3 (4 cos θ − 4 + θ sin θ)
• P = 2 :
b0 = −2sin3 θ
θ2 (θ cos θ − 3 sin θ)
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Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula : coefficientsE.g. b0 in case M = 6As a series . . .
• P = −1 :b0 = 1
• P = 0 :
b0 = 1 −16θ2 +
180
θ4 + O(θ6)
• P = 1 :
b0 = 1 −13θ2 +
37720
θ4 + O(θ6)
• P = 2 :
b0 = 1 −12θ2 +
760
θ4 + O(θ6)
![Page 51: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/51.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula : local truncation error
lte = L[y ](t)
As an inifinite series :
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2)
In closed form : (Coleman and Ixaru)
lte = hM ΦK ,P(Z ) DK+1 (D2 + ω2)P+1y(ξ)
Z ∈ some interval ΦK ,P(0) 6= 0 ξ ∈ (t − 2 h, t + 2 h)
![Page 52: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/52.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula : local truncation error
lte = L[y ](t)
As an inifinite series :
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2)
In closed form : (Coleman and Ixaru)
lte = hM ΦK ,P(Z ) DK+1 (D2 + ω2)P+1y(ξ)
Z ∈ some interval ΦK ,P(0) 6= 0 ξ ∈ (t − 2 h, t + 2 h)
![Page 53: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/53.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Central formula : local truncation error
lte = L[y ](t)
As an inifinite series :
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2)
In closed form : (Coleman and Ixaru)
lte = hM ΦK ,P(Z ) DK+1 (D2 + ω2)P+1y(ξ)
Z ∈ some interval ΦK ,P(0) 6= 0 ξ ∈ (t − 2 h, t + 2 h)
![Page 54: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/54.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
• P = 0 :y (K+3)(tj) + y (K+1)(tj)ω2
j = 0
• P = 1 :
y (K+5)(tj) + 2 y (K+3)(tj)ω2j + y (K+1)(tj)ω4
j = 0
• P = 2 :
y (K+7)(tj)+3 y (K+5)(tj)ω4j +3 y (K+3)(tj)ω4
j +y (K+1)(tj)ω6j = 0
• . . .
![Page 55: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/55.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
• P = 0 :y (K+3)(tj) + y (K+1)(tj)ω2
j = 0
• P = 1 :
y (K+5)(tj) + 2 y (K+3)(tj)ω2j + y (K+1)(tj)ω4
j = 0
• P = 2 :
y (K+7)(tj)+3 y (K+5)(tj)ω4j +3 y (K+3)(tj)ω4
j +y (K+1)(tj)ω6j = 0
• . . .
![Page 56: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/56.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
• P = 0 :y (K+3)(tj) + y (K+1)(tj)ω2
j = 0
• P = 1 :
y (K+5)(tj) + 2 y (K+3)(tj)ω2j + y (K+1)(tj)ω4
j = 0
• P = 2 :
y (K+7)(tj)+3 y (K+5)(tj)ω4j +3 y (K+3)(tj)ω4
j +y (K+1)(tj)ω6j = 0
• . . .
![Page 57: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/57.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
• P = 0 :y (K+3)(tj) + y (K+1)(tj)ω2
j = 0
• P = 1 :
y (K+5)(tj) + 2 y (K+3)(tj)ω2j + y (K+1)(tj)ω4
j = 0
• P = 2 :
y (K+7)(tj)+3 y (K+5)(tj)ω4j +3 y (K+3)(tj)ω4
j +y (K+1)(tj)ω6j = 0
• . . .
![Page 58: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/58.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
• P = 0 :y (K+3)(tj) + y (K+1)(tj)ω2
j = 0
• P = 1 :
y (K+5)(tj) + 2 y (K+3)(tj)ω2j + y (K+1)(tj)ω4
j = 0
• P = 2 :
y (K+7)(tj)+3 y (K+5)(tj)ω4j +3 y (K+3)(tj)ω4
j +y (K+1)(tj)ω6j = 0
• . . .
![Page 59: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/59.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
ω2j is solution of equation of degree P + 1.
• Which value of P should be chosen ?
• Which root ωj should be chosen ?
![Page 60: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/60.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
ω2j is solution of equation of degree P + 1.
• Which value of P should be chosen ?
• Which root ωj should be chosen ?
![Page 61: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/61.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Local truncation error
lte = hM CM DK+1 (D2 + ω2)P+1y(t) + O(hM+2) ,
At tj : D(K+1) (D2 + ω2j )(P+1)y(t)
∣
∣
∣
t=tj= 0 j = 2, . . . , N − 1
ω2j is solution of equation of degree P + 1.
• Which value of P should be chosen ?
• Which root ωj should be chosen ?
![Page 62: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/62.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
lte = hM CM DK+1 (D2 − µ2)P+1y(t) + O(hM−2)
Suppose y(t) takes the form tP0 eµ0 t
Then lte= 0 for any EF rule with P ≥ P0 and µj = µ0
TheoremIf y(t) = tP0 eµ0 t then ν = µ2
0 is a root of multiplicity P − P0 + 1of DK+1 (D2 − ν)P+1y(t) = 0.
• if P = P0, then µ = µ0 will be a single root
• if P = P0 + 1, then µ = µ0 will be a double root
• if P = P0 + 2, then µ = µ0 will be a triple root
• . . .
![Page 63: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/63.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
lte = hM CM DK+1 (D2 − µ2)P+1y(t) + O(hM−2)
Suppose y(t) takes the form tP0 eµ0 t
Then lte= 0 for any EF rule with P ≥ P0 and µj = µ0
TheoremIf y(t) = tP0 eµ0 t then ν = µ2
0 is a root of multiplicity P − P0 + 1of DK+1 (D2 − ν)P+1y(t) = 0.
• if P = P0, then µ = µ0 will be a single root
• if P = P0 + 1, then µ = µ0 will be a double root
• if P = P0 + 2, then µ = µ0 will be a triple root
• . . .
![Page 64: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/64.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
lte = hM CM DK+1 (D2 − µ2)P+1y(t) + O(hM−2)
Suppose y(t) takes the form tP0 eµ0 t
Then lte= 0 for any EF rule with P ≥ P0 and µj = µ0
TheoremIf y(t) = tP0 eµ0 t then ν = µ2
0 is a root of multiplicity P − P0 + 1of DK+1 (D2 − ν)P+1y(t) = 0.
• if P = P0, then µ = µ0 will be a single root
• if P = P0 + 1, then µ = µ0 will be a double root
• if P = P0 + 2, then µ = µ0 will be a triple root
• . . .
![Page 65: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/65.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
lte = hM CM DK+1 (D2 − µ2)P+1y(t) + O(hM−2)
Suppose y(t) takes the form tP0 eµ0 t
Then lte= 0 for any EF rule with P ≥ P0 and µj = µ0
TheoremIf y(t) = tP0 eµ0 t then ν = µ2
0 is a root of multiplicity P − P0 + 1of DK+1 (D2 − ν)P+1y(t) = 0.
• if P = P0, then µ = µ0 will be a single root
• if P = P0 + 1, then µ = µ0 will be a double root
• if P = P0 + 2, then µ = µ0 will be a triple root
• . . .
![Page 66: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/66.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
Suppose y(t) does not take the form tP0 eµ0 t
Then y(t) 6∈ S for any P.
For a given value of P :
D(K+1) (D2 − µ2j )
(P+1)y(t)∣
∣
∣
t=tj= 0
At each point tj , this gives P + 1 values for µ2j .
Idea : keep |µj h| as small as possible.If possible, choose P ≥ 1 to avoid too large values for |µj |.
![Page 67: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/67.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
Suppose y(t) does not take the form tP0 eµ0 t
Then y(t) 6∈ S for any P.
For a given value of P :
D(K+1) (D2 − µ2j )
(P+1)y(t)∣
∣
∣
t=tj= 0
At each point tj , this gives P + 1 values for µ2j .
Idea : keep |µj h| as small as possible.If possible, choose P ≥ 1 to avoid too large values for |µj |.
![Page 68: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/68.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
Suppose y(t) does not take the form tP0 eµ0 t
Then y(t) 6∈ S for any P.
For a given value of P :
D(K+1) (D2 − µ2j )
(P+1)y(t)∣
∣
∣
t=tj= 0
At each point tj , this gives P + 1 values for µ2j .
Idea : keep |µj h| as small as possible.If possible, choose P ≥ 1 to avoid too large values for |µj |.
![Page 69: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/69.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
Suppose y(t) does not take the form tP0 eµ0 t
Then y(t) 6∈ S for any P.
For a given value of P :
D(K+1) (D2 − µ2j )
(P+1)y(t)∣
∣
∣
t=tj= 0
At each point tj , this gives P + 1 values for µ2j .
Idea : keep |µj h| as small as possible.If possible, choose P ≥ 1 to avoid too large values for |µj |.
![Page 70: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/70.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Parameter selection
Suppose y(t) does not take the form tP0 eµ0 t
Then y(t) 6∈ S for any P.
For a given value of P :
D(K+1) (D2 − µ2j )
(P+1)y(t)∣
∣
∣
t=tj= 0
At each point tj , this gives P + 1 values for µ2j .
Idea : keep |µj h| as small as possible.If possible, choose P ≥ 1 to avoid too large values for |µj |.
![Page 71: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/71.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
First example
y (4) −384 t4
(2 + t2)4 y = 242 − 11 t2
(2 + t2)4
y(−1) =13
y(1) =13
y ′′(−1) =2
27y ′′(1) =
227
Solution : y(t) =1
2 + t2
![Page 72: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/72.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
First example
y (4) −384 t4
(2 + t2)4 y = 242 − 11 t2
(2 + t2)4
y(−1) =13
y(1) =13
y ′′(−1) =2
27y ′′(1) =
227
Solution : y(t) =1
2 + t2
![Page 73: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/73.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 0 : y (8)(tj) − y (6)(tj)µ2j = 0
• re-express higher orderderivatives in terms of y ,y ′, y ′′ and y ′′′
• approximate y ′, y ′′ andy ′′′ in terms of y
• an initial approximationfor y can be computedwith a polynomial rule
![Page 74: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/74.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 0 : y (8)(tj) − y (6)(tj)µ2j = 0
• re-express higher orderderivatives in terms of y ,y ′, y ′′ and y ′′′
• approximate y ′, y ′′ andy ′′′ in terms of y
• an initial approximationfor y can be computedwith a polynomial rule
![Page 75: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/75.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 0 : y (8)(tj) − y (6)(tj)µ2j = 0
• re-express higher orderderivatives in terms of y ,y ′, y ′′ and y ′′′
• approximate y ′, y ′′ andy ′′′ in terms of y
• an initial approximationfor y can be computedwith a polynomial rule
![Page 76: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/76.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 0 : y (8)(tj) − y (6)(tj)µ2j = 0
• re-express higher orderderivatives in terms of y ,y ′, y ′′ and y ′′′
• approximate y ′, y ′′ andy ′′′ in terms of y
• an initial approximationfor y can be computedwith a polynomial rule
![Page 77: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/77.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 0 : y (8)(tj) − y (6)(tj)µ2j = 0
• re-express higher orderderivatives in terms of y ,y ′, y ′′ and y ′′′
• approximate y ′, y ′′ andy ′′′ in terms of y
• an initial approximationfor y can be computedwith a polynomial rule
−1 −0.5 0 0.5 1
0
1
2
3
4
5
6
7
8
9
Real and imaginary partof µj
![Page 78: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/78.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 1 : y (8)(tj) − 2 y (6)(tj)µ2j + y (4)(tj)µ4
j = 0
−1 −0,5 0 0,5 1−5
0
5
10
−1 −0,5 0 0,5 1−5
0
5
10
Real and imag. part of µ1,j and µ2,j
![Page 79: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/79.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 1 : y (8)(tj) − 2 y (6)(tj)µ2j + y (4)(tj)µ4
j = 0
−1 −0,5 0 0,5 1−5
0
5
10
−1 −0,5 0 0,5 1−5
0
5
10
Real and imag. part of µ1,j and µ2,j
−1 −0,5 0 0,5 1
−4
−2
0
2
4
6
8
10
12
Real and imaginarypart of µj with smallestnorm
![Page 80: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/80.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 8
P = 1 : y (8)(tj) − 2 y (6)(tj)µ2j + y (4)(tj)µ4
j = 0
10−3
10−2
10−1
10−10
10−9
10−8
10−7
10−6
h
E
error obtained with µ1,j , µ2,j and µ with smallest norm
![Page 81: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/81.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Global error
M = 6 :(K , P) = (5,−1) : second-order method(K , P) = (1, 1) : fourth-order method
10−3
10−2
10−1
10−10
10−8
10−6
10−4
10−2
h
E
![Page 82: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/82.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Global error
M = 8 :(K , P) = (7,−1) : fourth-order method(K , P) = (3, 1) : sixth-order method
10−3
10−2
10−1
10−12
10−10
10−8
10−6
10−4
h
E
![Page 83: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/83.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Global error
M = 10 :(K , P) = (9,−1) : sixth-order method(K , P) = (5, 1) : eighth-order method
10−3
10−2
10−1
10−12
10−10
10−8
10−6
h
E
![Page 84: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/84.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Condition number
10−3
10−2
10−1
104
106
108
1010
1012
h
cond
(A)
![Page 85: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/85.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Second example
y (4) − t = 4 et
y(−1) = −1/e y(1) = e
y ′′(−1) = 1/e y ′′(1) = 3 e
Solution : y(t) = et t
![Page 86: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/86.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Second example
y (4) − t = 4 et
y(−1) = −1/e y(1) = e
y ′′(−1) = 1/e y ′′(1) = 3 e
Solution : y(t) = et t
![Page 87: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/87.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by fourth-order finite difference scheme
![Page 88: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/88.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by fourth-order finite difference scheme
![Page 89: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/89.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by fourth-order finite difference scheme
![Page 90: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/90.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by fourth-order finite difference scheme
−1 −0.5 0 0.5 10
0.5
1
1.5
2
t
µ j
two real roots µ(1) and µ(2)
![Page 91: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/91.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
µj for M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by fourth-order finite difference scheme
−1 −0.5 0 0.5 10
0.5
1
1.5
2
t
µ j
two real roots µ(1) and µ(2)10
−310
−210
−1
10−8
10−7
10−6
10−5
10−4
10−3
10−2
h
E
![Page 92: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/92.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by sixth-order finite difference scheme
−1 −0.5 0 0.5 10
0.5
1
1.5
2
t
µ j
two real roots µ(1) and µ(2)
![Page 93: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/93.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
M = 6
P = 1 : y (6)(tj) − 2 y (4)(tj)µ2j + y (2)(tj)µ4
j = 0
differentiating the differential equation :
(y (2)(tj) + 4 etj ) − 2 (yj + 4 etj )µ2j + y (2)(tj)µ4
j = 0
y (2)(tj) approximated by sixth-order finite difference scheme
−1 −0.5 0 0.5 10
0.5
1
1.5
2
t
µ j
two real roots µ(1) and µ(2)10
−310
−210
−110
−14
10−12
10−10
10−8
10−6
10−4
10−2
h
E
![Page 94: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/94.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Conclusions
• Fourth-order boundary value problems are solved bymeans of parameterized exponentially-fitted methods.
• A suitable value for the parameter can be found from theroots of the leading term of the local truncation error.
• If a constant value is found, then a very accurate solutioncan be obtained.
• However, the methods strongly suffer from the fact that thesystem to be solved is ill-conditioned for small values of themesh size.
![Page 95: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/95.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Conclusions
• Fourth-order boundary value problems are solved bymeans of parameterized exponentially-fitted methods.
• A suitable value for the parameter can be found from theroots of the leading term of the local truncation error.
• If a constant value is found, then a very accurate solutioncan be obtained.
• However, the methods strongly suffer from the fact that thesystem to be solved is ill-conditioned for small values of themesh size.
![Page 96: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/96.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Conclusions
• Fourth-order boundary value problems are solved bymeans of parameterized exponentially-fitted methods.
• A suitable value for the parameter can be found from theroots of the leading term of the local truncation error.
• If a constant value is found, then a very accurate solutioncan be obtained.
• However, the methods strongly suffer from the fact that thesystem to be solved is ill-conditioned for small values of themesh size.
![Page 97: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/97.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Conclusions
• Fourth-order boundary value problems are solved bymeans of parameterized exponentially-fitted methods.
• A suitable value for the parameter can be found from theroots of the leading term of the local truncation error.
• If a constant value is found, then a very accurate solutioncan be obtained.
• However, the methods strongly suffer from the fact that thesystem to be solved is ill-conditioned for small values of themesh size.
![Page 98: Exponentially-fitted methods applied to fourth-order boundary ...mvdaele/voordrachten/kos2008.pdf• linear multistep methods • Runge-Kutta methods Aim : build methods which perform](https://reader036.vdocument.in/reader036/viewer/2022081403/608e36bcc4ee4176a25493ea/html5/thumbnails/98.jpg)
Introduction Fourth-order boundary value problems Numerical examples Conclusions
Conclusions
• Fourth-order boundary value problems are solved bymeans of parameterized exponentially-fitted methods.
• A suitable value for the parameter can be found from theroots of the leading term of the local truncation error.
• If a constant value is found, then a very accurate solutioncan be obtained.
• However, the methods strongly suffer from the fact that thesystem to be solved is ill-conditioned for small values of themesh size.