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J. Korean Soc. Ind. Appl. Math. Vol.23, No.3, 253–266, 2019 http://doi.org/10.12941/jksiam.2019.23.253

SYSTEMATIC APPROXIMATION OF THREE DIMENSIONAL FRACTIONALPARTIAL DIFFERENTIAL EQUATIONS IN FLUID MECHANICS

FIRDOUS KHAN1† AND KIRTIWANT P. GHADLE1

1DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY,AURANGABAD-431004 (M.S.) INDIA

Email address: [email protected], [email protected]

ABSTRACT. In this article, a systematic solution based on the sequence of expansion methodis planned to solve the time-fractional diffusion equation, time-fractional telegraphic equationand time-fractional wave equation in three dimensions using a current and valid approximatemethod, namely the ADM, VIM, and the NIM subject to the estimate initial condition. By usingthese three methods it is likely to find the exact solutions or a nearby approximate solutionof fractional partial differential equations. The exactness, efficiency, and convergence of themethod are demonstrated through the three numerical examples.

1. INTRODUCTION

Many real phenomena emerging in the engineering and science field can be verified ef-fectively by building model, with the fractional calculus theory. Partial fractional differentialtheory has improved much more interest as the fractional differential theory and fractional-order system response finally converges to the integer-order equations. The fractional differ-entiation applied in mathematical modeling, the traffic flow model with fractional derivatives,measurement of viscoelastic material properties, etc., have been affecting a large area in thismodern era. Before the nineteenth century, no systematic solution method was offered forsuch type of equations even for the linear fractional partial differential equations. In recentyears, from the literature, it can be seen that much attention has been given to the developmentof systematic and numerical schemes for the one and two dimensional hyperbolic fractionaland non-fractional time-fractional telegraphic equation [1] and differential equations with frac-tional order have proved to be valuable tools to the modeling of many physical phenomena[2]. Comparison of the ADM, VIM, and NIM in one dimension equations is done by Ghadleand Khan [3]. VIM of approximation solution of nonlinear Volterra-Fredholm integral equa-tions (VFIEs) of the second type is done by [4]. Some approximation methods like ADM,

Received by the editors August 8 2019; Revised September 12 2019; Accepted in revised form September 142019; Published online September 25 2019.

2000 Mathematics Subject Classification. 35G40, 35R11, 76M25.Key words and phrases. Fractional partial differential, Adomian decomposition method (ADM), Variation iter-

ation method (VIM), New iterative method (NIM).† Corresponding author.

253

254 FIRDOUS KHAN AND KIRTIWANT P. GHADLE

VIM and Homotopy analysis method (HAM) are used to solve fuzzy VFIEs [5]. The modifiedADM is used to solve VFIEs and has also found its uniqueness and existence [6]. The Homo-topy perturbation method is applied to Caputo fractional Volterra-Fredholm integro-differentialequations (VFIDEs) to find the approximate result [7]. Using HAM the author had convertedfuzzy VFIDEs into crisp case [8]. The author applied ADM and modified Laplace ADM tofind the results of nonlinear fractional VFIDEs [9]. ADM applied to the Fuzzy VFIEs of thefirst and second kind and convert it into the crisp case is done by [10].

Here we are taking three-dimension linear time fractions equations. The first example weencountered is the diffusion equation which has commenced in physics by Nigmatullin to aparticular form of porous media to explain diffusion in media with fractal geometry. The frac-tured porous medium has an incredibly complex structure, which can be measured as a fractal.Hence, the material particles while drifting next to fractures and pore channels will carry outcomposite motion compelled by the complex geometry of the pores and fractures and their al-location in the domain. Therefore, the position of the force field in this situation will be takenby the porous channels and stochastically distributed fractures. Based on the resemblance ofthe aforementioned processes it can assume that the diffusion equation in the porous mediumwill be comparable to the equations of irregular diffusion validated for the case of basic par-ticles motion beneath the outcome of different force fields [11]. The second example is thetelegraphic equation which is used in the study of signal analysis for transmission and prop-agation of electrical signals, modeling reaction-diffusion in a different branch of engineeringscience in the propagation of pressure waves in the study of pulsatile blood flow in the arteries.The third example is the wave equation which mainly arises in mechanical waves such as soundwave, water wave, and seismic wave i.e. the waves which are travel through layers of earth andis the result of earthquakes, large landslides, and volcano eruption. It arises in fluid dynamics,electromagnetic and acoustics fields.

In this paper, we use a systematic scheme, namely the ADM, VIM and the NIM based onthe series solution method to find the solutions of the time-fractional diffusion equation, time-fractional telegraphic equation and the time fraction wave equation in three dimensions. Theexactness and efficiency of the proposed method are demonstrated by the three examples. Themain advantage of the method is that it solves the equations directly without using linearization,transformation, discretization or restrictive assumptions.

2. PRELIMINARIES

A few essential definitions and properties of the fractional calculus theory which are neces-sary for this paper are given in this section.

Definition 1. A real function f(t), t > 0, is said to be in the space Cµ, µ ∈ R if ∃ aR p(> µ), s.t f(t) = tpf1(t), where f1(t) ∈ C[0, ∞), and it is said to be in the space Cm

µ

iff f (m) ∈ Cµ, m ∈ N [12].

SYSTEMATIC APPROXIMATION OF THREE DIMENSIONAL 255

Definition 2. The Riemann-Liouville fractional integral operator of order α ≥ 0 of a functionf ∈ Cµ, µ ≥ −1, is defined as

Iαf(t) =1

Γ(α)

∫(t− τ)α−1f(τ)dτ, α > 0, t > 0, I0f(t) = f(t).

Properties of the operators Iα can be found in [13, 14]; we mention only the following: forf ∈ Cµ, µ ≥ −1, α, β ≥ 0 and γ > −1,

(1) IαIβf(t) = Iα+βf(t),(2) IαIβf(t) = IβIαf(t),(3) Iαtγ = Γ(γ+1)

Γ(γ+1+α) tα+γ .

The Riemann-Liouville derivative has some drawback when dealing with fractional differ-ential equations. Therefore, [15] initiate a modified fractional differential operator Dα.

Definition 3. The fractional derivative of f(t) in the Caputo sense is defined as [12]

Dαf(t) =I(m−α)D(m)f(t) =1

Γ(m− α)

∫ t

0(t− τ)m−α−1f (m)(t)dt,

for m− 1 < α ≤ m, m ∈ N, t > 0, f ∈ Cm−1.

Lemma 1. If m− 1 < α ≤ m, m ∈ N and f ∈ Cmµ , µ ≥ −1, then

DαIαf(t) = f(t),

IαDαf(t) = f(t)−m−1∑k=0

fk(0+)tk

k!, t > 0.

The Caputo fractional derivative is in use at this point for the reason that it permits conven-tional initial and boundary conditions to be integrated into the formulation of the difficulty[16]. It imitates on the one-dimensional linear inhomogeneous fractional partial differentialequations in fluid mechanics, where the unknown function u(x, t) is known to be a related func-tion of time, i.e, vanishing for t < 0. The fractional derivative is used in Caputo sense asfollows.

Definition 4. For m to be the smallest integer that exceeds α, the Caputo time-fractional de-rivative operator of order α > 0 is defined as [12]

Dαt u(x, t) =

∂αu(x, t)

∂tα

1

Γ(m−α)

∫ t0 (t− τ)m−α−1 ∂

mu(x,τ)∂τm dτ,

for m− 1 < α < m∂m

∂tmu(x, t),for α = m, m ∈ N.

3. MAIN RESULT

The three linear fractional partial differential equations viz. ADM [17], VIM [17], and NIM[18] are used to construct the solution of the given examples which has been discussed in thissection.


Example 1. Consider the following three-dimensional linear time-fractional diffusion equa-tion

∂αu

∂tα=

∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2, t > 0, x, y, z ∈ R, 0 < α ≤ 1.

Subject to the initial condition

u(x, y, z, 0) = sinx cos y cos z.

According to the ADM

uj+1(x, y, z, t) =− Jα(a0(x, y, z)uj(x, y, z, t) + a1(x, y, z)L1x,y,zuj(x, y, z, t)

+ · · ·+ an(x, y, z)Lnx,y,zuj(x, y, z, t)). (3.1)

we obtain the recurrence relation in view of (3.1)

u0(x, y, z, t) = sinx cos y cos z,

u1(x, y, z, t) =3tα

Γ(α+ 1)sinx cos y cos z,

u2(x, y, z, t) =9t2α

Γ(2α+ 1)sinx cos y cos z,

u3(x, y, z, t) =27t3α


the series form is given by

u(x, y, z, t) =[1 +

3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)+

27t3α

Γ(3α+ 1)+ · · ·

]sinx cos y cos z.

Now according to the VIM and its formula is given by

uk+1(x, y, z, t) = uk(x, y, z, t) + Jαt

[ ∂α

∂tαuk −

∂2

∂x2uk −

∂2

∂y2uk −

∂2

∂z2uk

]. (3.2)

now obtaining the recurrence relation in view of (3.2)


u1(x, y, z, t) =(1 +3tα

Γ(α+ 1)) sinx cos y cos z,

u2(x, y, z, t) =(1 +3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)) sinx cos y cos z,

u3(x, y, z, t) =(1 +3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)+

27t3α

Γ(3α+ 1)) sinx cos y cos z,

...


Now according to the NIM and by the formula

u(x, y, z, t) =

m−1∑k=0

hk(x, y, z)tk

k!+ Iαt B + Iαt A = f +N(u)

where f =m−1∑k=0

hk(x, y, z)tk

k!+ Iαt B and N(u) = Iαt A and u0 = f,

un+1 =N(un), n = 0, 1, 2, · · ·

where

u0(x, y, z, t) = sinx cos y cos z.

Let N(u) = −Jαt [u

2x + u2y + u2z] we can obtain the following first few components of the new

iterative solution,


u1(x, y, z, t) =3tα

Γ(α+ 1)sinx cos y cos z,

u2(x, y, z, t) =9t2α


u3(x, y, z, t) =27t3α


The series form is given by

u(x, y, z, t) =[1 +

3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)+

27t3α

Γ(3α+ 1)+ · · ·

]sinx cos y cos z. (3.3)

Example 2. Consider the following three-dimensional linear time-fractional telegraphic equa-tion (TFTE).

∂2αu

∂t2α+ 2

∂αu

∂tα+ u =

∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2, t > 0, x, y, z ∈ R, 0 < α ≤ 1. (3.4)

subject to the initial conditions

u(x, y, z, 0) = sinh(x) sinh(y) sinh(z),

ut(x, y, z, 0) = − sinh(x) sinh(y) sinh(z).

In the view of the ADM the firs few components of above problem are derived as follows,





u0(x, y, z, t) =(1− t− 4tα

Γ(α+ 1)) sinh(x) sinh(y) sinh(z),

u1(x, y, z, t) =(− 4tα

Γ(α+ 1)+

4tα+1

Γ(α+ 2)+

16t2α

Γ(2α+ 1)) sinh(x) sinh(y) sinh(z),

u2(x, y, z, t) =(16t2α

Γ(2α+ 1)− 16t2α+1

Γ(2α+ 2)− 64t3α


...


u(x, y, z, t) =(1− t− 8tα

Γ(α+ 1)+

4tα+1

Γ(α+ 2)+

32t2α

Γ(2α+ 1)− 16t2α+1

Γ(2α+ 2)

− 64t3α

Γ(3α+ 1)+ · · ·

)sinh(x) sinh(y) sinh(z). (3.6)

Now according to the VIM, the iteration formula for this problem is given by,

uk+1(x, y, z, t) =uk(x, y, z, t)− (α− 1)Iαt

[∂2αuk∂t2α

+ 2∂αuk∂tα

+ uk −∂2uk∂x2

− ∂2uk∂y2

− ∂2uk∂z2

].

By the above iteration formula, if we began with u0 = sinh(x) sinh(y) sinh(z), we can obtainthe following approximation

u1(x, y, z, t) =(1− (α− 1)4tα


u2(x, y, z, t) =(1− (α− 1)4tα

Γ(α+ 1)− (α− 1)

4tα

Γ(α+ 1)+ (α− 1)2

4tα+1

Γ(α+ 2)

+ (α− 1)216t2α


u3(x, y, z, t) =(1− (α− 1)4tα

Γ(α+ 1)− (α− 1)

4tα

Γ(α+ 1)+ (α− 1)2

4tα+1

Γ(α+ 2)

+ (α− 1)216t2α

Γ(2α+ 1)+ (α− 1)2

16t2α

Γ(2α+ 1)− (α− 1)3

16t2α+1

Γ(2α+ 2)

− (α− 1)364t3α


...



u(x, y, z, t) =(1− (α− 1)4tα

Γ(α+ 1)− (α− 1)

4tα

Γ(α+ 1)+ (α− 1)2

4tα+1

Γ(α+ 2)

+ (α− 1)216t2α

Γ(2α+ 1)+ (α− 1)2

16t2α

Γ(2α+ 1)− (α− 1)3

16t2α+1

Γ(2α+ 2)

− (α− 1)364t3α

Γ(3α+ 1)) sinh(x) sinh(y) sinh(z) · · · , (3.7)

According to the NIM and by the formula

u(x, y, z, t) =

m−1∑k=0

hk(x, y, z)tk


where f =m−1∑k=0

hk(x, y, z)tk


un+1 =N(un), n = 0, 1, 2, · · ·

Let N(u) = −Iαt [u2x + u2y +

2z −u] we can obtain the following first few components of the new

iterative solution

u0(x, y, z, t) =(1− t− 4tα


u1(x, y, z, t) =(− 4tα

Γ(α+ 1)+

4tα+1

Γ(α+ 2)+

16t2α


u2(x, y, z, t) =(16t2α

Γ(2α+ 1)− 16t2α+1

Γ(2α+ 2)− 64t3α


...


u(x, y, z, t) =(1− t− 8tα

Γ(α+ 1)+

4tα+1

Γ(α+ 2)+

32t2α

Γ(2α+ 1)− 16t2α+1

Γ(2α+ 2)

− 64t3α

Γ(3α+ 1)+ · · ·

)sinh(x) sinh(y) sinh(z). (3.8)

From (3.6), (3.7) and (3.8), the ADM, the VIM and the NIM gives the same solution for thethree-Dimensional linear time-fractional telegraphic equation (3.4) (when α = 2).

Example 3. Consider the following three-dimensional linear time-fractional wave equation.

∂αu

∂tα=

∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2, t > 0, x, y, z ∈ R, 0 < α ≤ 1.


Subject to the initial condition

u(x, y, z, 0) = sinx cos y cot z.

According to the ADM




u0(x, y, z, t) =(1− 3tα

Γ(α+ 1)) sinx cos y cot z,

u1(x, y, z, t) =(− 3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)) sinx cos y cot z,

u2(x, y, z, t) =(9t2α

Γ(2α+ 1)− 27t3α


u3(x, y, z, t) =(− 27t3α

Γ(3α+ 1)+

81t4α


...

the series form is given by Now according to the VIM and its formula is given by

uk+1(x, y, z, t) = uk(x, y, z, t)− Jαt

[ ∂α

∂tαuk −

∂2

∂x2uk −

∂2

∂y2uk −

∂2

∂z2uk

]. (3.10)

now obtaining the recurrence relation in view of (3.10)

u0(x, y, z, t) =(1− 3tα


u1(x, y, z, t) =(1− 3tα

Γ(α+ 1)− 3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)) sinx cos y cot,

u2(x, y, z, t) =(1− 3tα

Γ(α+ 1)− 3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)+

9t2α

Γ(2α+ 1)

− 27t3α


u3(x, y, z, t) =(1− 3tα

Γ(α+ 1)− 3tα

Γ(α+ 1)+

9t2α

Γ(2α+ 1)+

9t2α

Γ(2α+ 1)

− 27t3α

Γ(3α+ 1)− 27t3α

Γ(3α+ 1)+

81t4α


...


Now according to the NIM and by the formula

u(x, y, z, t) =

m−1∑k=0

hk(x, y, z)tk


where f =

m−1∑k=0

hk(x, y, z)tk


un+1 =N(un), n = 0, 1, 2, · · ·where

u0(x, y, z, t) = sinx cos y cot z.

Let N(u) = u+Jαt [u

2x+u2y +u2z] we can obtain the following first few components of the new

iterative solution,

u0(x, y, z, t) =(1− 3tα


u1(x, y, z, t) =(− 3tα

Γ(α+ 1)+

9t2α


u2(x, y, z, t) =(9t2α

Γ(2α+ 1)− 27t3α


u3(x, y, z, t) =(− 27t3α

Γ(3α+ 1)+

81t4α


...

the series form is given by

u(x, y, z, t) =[1− 6tα

Γ(α+ 1)+

18t2α

Γ(2α+ 1)− 54t3α

Γ(3α+ 1)+

81t4α

Γ(4α+ 1)+ · · ·

]sinx cos y cot z. (3.11)

Figure 1(a), 2(a), and 3(a) demonstrate the numerical approximation figure for Time frac-tional diffusion equation, time fractional telegraphic equation, and time fractional wave equa-tion with NIM, whereas figure 1(b), 2(b), and 3(b) demonstrate the exact figures at t = 0.2 andx = 3 in figure1(b) and t = 0.2 and x = 2 in figure 2(b) and 3(b). Table 1, 2, and 3 shows thenumerical solution at α = 0.25, α = 0.5, α = 0.75, and α = 1, exact solution and absoluteerror of NIM.


TABLE 1. Time fractional diffusion equation (using equation(3.3)) when α =0.5, α = 0.25, α = 0.75 and α = 1 and absolute error for α = 1.

t x α = 0.5 α = 0.25 α = 0.75 α = 1 Exact Error0.2 1 5.158667 13.918794 2.410888 1.528111 0.673177 0.854934

2 5.574479 15.040713 2.605216 1.651284 0.727438 0.9238463 0.865141 2.3342698 0.404321 0.256273 0.112896 0.1433774 -4.63960 -12.51829 -2.16830 -1.37435 -0.605442 -0.7689085 -5.87871 -15.86159 -2.74740 -1.74140 -0.767139 -0.974261

0.3 1 7.481960 17.603940 3.484619 2.041829 0.58903 1.4527992 8.085041 19.022899 3.765496 2.206410 0.636508 1.5699023 1.254772 2.9522921 0.584392 0.342427 0.098784 0.2436434 -6.72912 -15.83263 -3.13399 -3.13399 -0.529762 -2.6004225 -8.52629 -20.06111 -3.97100 -2.32682 -0.671247 -1.655573

0.4 1 9.996024 20.894781 4.798267 2.699438 0.504883 2.1945552 10.80174 22.578997 5.185029 2.917026 0.545578 2.3714483 1.676396 3.5041870 0.804699 0.452712 0.084672 0.368044 -8.99022 -18.79235 -4.31546 -2.42782 -0.454081 -1.9737395 -11.3912 -27.26439 -7.25052 -3.07622 -0.575355 2.500865

FIGURE 1. Time fractional diffusion equation (using equation(3.3)) (a) whenα = 1, t=0.2 and x=3 and (b) is the exact solution graph at t=0.2 and x=3.


TABLE 2. Time fractional telegraphic equation (using equation(3.8)) whenα = 0.5, α = 0.25, α = 0.75 and α = 1 and absolute error at α = 1.

t x α = 0.5 α = 0.25 α = 0.75 α = 1 Exact Error0.2 1 -1.403218 -12.204161 -0.304506 -0.219370 -0.23504 0.01567

2 -4.330558 -37.664012 -0.939755 -0.677013 -0.72537 0.0483593 -11.96158 -104.03305 -2.595730 -1.870003 -2.00357 0.1335674 -32.58481 -283.39876 -7.071088 -5.094117 -5.45798 0.3638635 -88.60041 -770.58124 -19.22678 -13.85126 -14.8406 0.98934

0.3 1 -3.267969 -18.187893 -0.564223 -0.517088 -0.35256 -0.1645282 -10.08548 -56.130773 -1.741285 -1.595818 -1.08806 -0.5077583 -27.85745 -155.04072 -4.809664 -4.407864 -3.00536 -1.4025044 -75.88710 -422.34990 -13.10211 -12.00756 -8.18698 -3.820585 -206.3423 -1148.3992 -35.62557 -32.64941 -22.261 -10.38841

0.4 1 -5.884927 -24.126100 -1.022009 -0.673782 -0.47008 -0.2037022 -18.16183 -74.457037 -3.154087 -2.079399 -1.45074 -0.6286593 -50.16542 -205.66032 -8.712011 -5.743581 -4.00715 -1.7364314 -136.6567 -560.24388 -23.73258 -15.64621 -10.916 -4.730215 -371.5793 -1523.3426 -64.53057 -42.54317 -29.6813 -12.86187

FIGURE 2. Time fractional telegraphic equation (using equation (3.8))(a)When α = 1, t=0.2 and x=2 and (b) is the exact solution graph at t=0.2 andx=2.


TABLE 3. Time fraction wave equation (using equation ) when α = 0.5, α =0.25, α = 0.75 and α = 1 and absolute error at α = 0.75.

t x α = 0.5 α = 0.25 α = 0.75 α = 1 Exact Error0.2 1 -0.371147 3.605282 -0.168267 0.078593 -0.168294 0.000027

2 -0.401063 3.895884 -0.181830 0.084928 -0.181859 0.0000293 -0.062243 0.604628 -0.028219 0.013180 -0.028224 0.0000054 0.3338027 -3.24252 0.1513365 -0.07068 0.15136 -0.0000235 0.4229525 -4.10851 0.1917544 -0.08956 0.191785 0.160938

0.3 1 -0.284459 6.486496 -0.393666 -0.17305 -0.252441 -0.1412252 -0.307388 7.009338 -0.425398 -0.18700 -0.272789 -0.1526093 -0.047705 1.087826 -0.066020 -0.02902 -0.042336 -0.0236844 0.2558375 -5.83382 0.3540560 0.155645 0.227041 0.1275195 0.3241648 -7.39188 0.4486150 0.197214 0.287677 0.160938

0.4 1 0.1022784 9.617066 -0.580142 -0.37832 -0.336588 -02435542 0.1105225 10.39224 -0.626904 -0.40882 -0.363719 -0.2631853 0.0171527 1.612842 -0.097293 -0.06344 -0.056448 -0.0408454 -0.091987 -8.64940 0.5217687 0.340258 0.302721 0.21904775 -0.116554 -10.9594 0.6611193 0.431132 0.38357 0.2775493

FIGURE 3. Time fractional wave equation (using equation (3.11)) (a) whenα = 0.75, t=0.2 and x=2 and (b) is the exact solution graph at t=0.2 and x=2.


4. CONCLUSION

In the current study, we have illustrated the ADM, VIM and the NIM for the systematic so-lution of the three-dimensional second-order hyperbolic linear time-fractional diffusion equa-tion, time-fractional telegraphic equation, and the time fraction wave equation. The method isapplied in a straightforward way without using linearization, translation assumptions. Theseresults show that the NIM techniques is highly exact, and quickly reach a stable endpoint andare very easily carried out mathematically for more than two-dimensional physical problemsrising in various field of engineering and sciences.

ACKNOWLEDGMENTS

The authors would like to thanks to Khan Arshiya Anjum for her Immense support and toDepartment of Mathematics, Dr. Babasaheb Ambedkar Marathwada University for infrastruc-ture.

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