wave equations in 1 dimension
DESCRIPTION
Document of ordinary differential equations, telling How to solve wave equation in 1 Dimension.TRANSCRIPT
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BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
1 Partial Differential Equations (PDEs)
Method of Separation of Variable (MSV) for Wave Equations
1. One-dimensional Wave Equation
The homogeneous partial differential equation that describes the vibrations of a vibrating string
will be discussed by using a well-known method called the method of separation of variables.
The most important feature of the method of separation of variables is that it reduces the partial
differential equation into a system of ordinary differential equations that can be easily handled.
1.1. Analysis of the Method:
The initial-boundary value problem that controls the vibrations of a string is given by
(1)
BCs (2)
IC (3)
The wave function is the displacement of any point of a vibrating string at position x at
time t. The method of separation of variables consists of assuming that the displacement
is identified as the product of two distinct functions and , where depends on the
space variable x and depends on the time variable t. This assumption allows us to set
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
The left hand side of Eq. (8) depends only on t and the right hand side depends only on x. This
means that the equality holds only if both sides are equal to the same constant. Therefore, we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. However, we can easily
show that selecting the constant to be zero or will produce the trivial solution
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
2 Partial Differential Equations (PDEs)
The result (9) gives two distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
where and are constants. To determine the constants and we use the homogeneous
boundary conditions
The condition gives
(14)
Similarly the condition gives
(15)
Now we solve Eq. (13) along with the conditions given in Eq. (14) and Eq. (15)
and
We exclude since it gives the trivial solution Accordingly, we find
It is important to note that is excluded since it gives the trivial solution The
function associated with is
Consequently, the solution associated with must satisfy
(16)
Proceeding as before the general solution of Eq. (16) is given by
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
3 Partial Differential Equations (PDEs)
where and are constants.
Combining the results and we obtain the infinite sequence of product functions
Recall that the superposition principle admits that a linear combination of the functions
also satisfies the given equation and the boundary conditions. Therefore, using this
principle gives the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
so that the constants can be determined in this case by using Fourier coefficients given by the
formula
To determine , we substitute in Eq. (18) and by using the initial condition
we obtain
(19)
so that
(20)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
4 Partial Differential Equations (PDEs)
Having determined the constants and , the particular solution follows immediately
upon substituting Eq. (19) and Eq. (20) into Eq. (17).
Note: If we have the ICs in term of sine or cosine function the we compare the coefficients of
sine or cosine function to fine the values of and
1.2. One-dimensional Wave Equation with Dirichlet BCs: In this section we
consider two types of Wave equation subject to different type of initial conditions, the general
form of these two types of problems given below:
Type 1:
BCs
ICs
Type 2:
BCs
ICs
Problem 1: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
5 Partial Differential Equations (PDEs)
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
Consequently, we have Eq. (14) is
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
6 Partial Differential Equations (PDEs)
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
To determine , we substitute in Eq. (18) and by using the initial condition
we obtain
and except
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
Verification:
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
7 Partial Differential Equations (PDEs)
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in Eq. (19) we have
for the Eq. (19), we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
Problem 2: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
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BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
8 Partial Differential Equations (PDEs)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
9 Partial Differential Equations (PDEs)
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
and except
To determine , we substitute in Eq. (18) and by using the initial condition
we obtain
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
10 Partial Differential Equations (PDEs)
Verification:
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in Eq. (19) we have
for the Eq. (19), we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
1.3. One-dimensional Wave Equation with Neumann BCs: In this section we
consider three types of Wave equation subject to different type of initial conditions, the general
form of these three types of problems given below:
Type 1:
BCs
ICs
Type 2:
BCs
ICs
Type 3:
BCs
ICs
Problem 1: Consider the One dimensional Wave equation subject to Neumann boundary
conditions as
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
11 Partial Differential Equations (PDEs)
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
12 Partial Differential Equations (PDEs)
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16), first we
differentiate Eq. (13), w.r.t we get
(13)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
13 Partial Differential Equations (PDEs)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
To determine , we substitute in Eq. (18) and by using the initial condition
we obtained
and except
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
Verification:
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in we have
for the expression we have
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
14 Partial Differential Equations (PDEs)
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
Problem 2: Consider the One dimensional Wave equation subject to Neumann boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
15 Partial Differential Equations (PDEs)
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16), first we
differentiate Eq. (13), w.r.t we get
(13)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
16 Partial Differential Equations (PDEs)
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
and except
To determine , we substitute in Eq. (18) and by using the initial condition
we obtained
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
Verification:
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
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BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
17 Partial Differential Equations (PDEs)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in we have
for the expression we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
Problem 3: Consider the One dimensional Wave equation subject to Neumann boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
18 Partial Differential Equations (PDEs)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16), first we
differentiate Eq. (13), w.r.t we get
(13)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
19 Partial Differential Equations (PDEs)
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
and except
To determine , we substitute in Eq. (18) and by using the initial condition
we obtained
except
Substituting the values of s and s into Eq. (17), we have
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
20 Partial Differential Equations (PDEs)
(19)
This is our required solution.
Verification:
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in we have
for the expression we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
1.4. One-dimensional Wave Equation with Dirichlet BCs (Use of Fourier
Coefficient): In this section we consider two types of Wave equation subject to different type
of initial conditions, the general form of these two types of problems given below:
Type 1:
BCs
ICs
Type 2:
BCs
ICs
where and are not trigonometric function in sine and cosine functions.
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
21 Partial Differential Equations (PDEs)
Problem 1: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
22 Partial Differential Equations (PDEs)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
23 Partial Differential Equations (PDEs)
The arbitrary constants are determined by using the Fourier coefficients method, therefore we
find
so that
This means that we can express by
To determine , we substitute in Eq. (18) and by using the initial condition
we obtain
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
Verification:
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in Eq. (19) we have
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
24 Partial Differential Equations (PDEs)
for the Eq. (19), we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
Problem 2: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(4)
Differentiating both sides of Eq. (4) twice with respect to t and twice with respect to x we obtain
(5)
(6)
Substituting Eq. (4) and Eq. (6) into Eq. (1) yields
(7)
Dividing both sides of Eq. (7) by gives
(8)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(9)
The selection of in Eq. (9) is essential to obtain nontrivial solutions. The result (9) gives two
distinct ordinary differential equations given by
(10)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
25 Partial Differential Equations (PDEs)
(11)
To determine the function , we solve the second order linear ODE
(12)
The auxiliary of Eq. (12) is
There the solutions of Eq. (12) is
(13)
Similarly we have
(14)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(15)
Similarly the condition gives
(16)
Now we solve Eq. (13) along with the conditions given in Eq. (15) and Eq. (16)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (13) is
Consequently, we have Eq. (14) is
Now, combining the results and we obtain the infinite sequence of product
functions
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
26 Partial Differential Equations (PDEs)
Using principle of super position the general solution by
(17)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (17)
with respect to is
(18)
To determine , we substitute in Eq. (17) and by using the initial condition
we obtain
To determine , we substitute in Eq. (18) and by using the initial condition
we obtain
The arbitrary constants are determined by using the Fourier coefficients method, therefore we
find
so that
Substituting the values of s and s into Eq. (17), we have
(19)
This is our required solution.
Verification:
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
27 Partial Differential Equations (PDEs)
Since Eq. (19) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (19) twice w.r.t and twice w.r.t we get
(20)
(21)
From Eq. (20) and Eq. (21) we observe that
Now for BCs substitute in Eq. (19) we have
for the Eq. (19), we have
Now for ICs substitute in Eq. (19) we have
putting into , we have
This implies that Eq. (19) is the solution of Eq. (1-3).
1.5. One-dimensional Wave Equation with Inhomogeneous Dirichlet BCs: In
this section we will consider the case where the boundary conditions of the vibrating string are
inhomogeneous. It is well known that the Method of Separation of Variables requires that the
equation and the boundary conditions are linear and homogeneous. Therefore, transformation
formulas should be used to convert the inhomogeneous boundary conditions to homogeneous
boundary conditions.
In this section we will discuss wave equations where Dirichlet boundary conditions are not
homogeneous.
In this first type of boundary conditions, the displacements and of a
vibrating string of length are given. We begin our analysis by considering the initial-boundary
value problem
(1)
BCs (2)
ICs (3)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
28 Partial Differential Equations (PDEs)
To convert the inhomogeneous boundary conditions of the Eq. (2) to homogeneous boundary
conditions, we simply use the the following transformation formula
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
(6)
Now differentiate twice Eq. (4) w.r.t we get
(7)
Eq. (6) and Eq. (7) implies that
For BCs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (4) at we get
For ICs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
BCs
ICs
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
29 Partial Differential Equations (PDEs)
In view of the above system, the method of separation of variables can be easily used in the
above system as discussed before. Having determined of the above system, the wave
function of Eq. (1-3) follows immediately upon substituting into Eq. (4).
Now, we consider two types of Wave equation subject to different type of initial conditions, the
general form of these two types of problems given below:
Type 1:
BCs
ICs
Type 2:
BCs
ICs
Note: Must use transformation in the end to convert the solution into
Problem 1: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
To convert the inhomogeneous BCs into homogenous BCs we use the following transformation
here we notice that
The above transformation reduces to
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
(6)
Now differentiate twice Eq. (4) w.r.t we get
(7)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
30 Partial Differential Equations (PDEs)
Eq. (6) and Eq. (7) implies that
For BCs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (4) at we get
For ICs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
(8)
BCs (9)
ICs (10)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(11)
Differentiating both sides of Eq. (11) twice with respect to t and twice with respect to x we
obtain
(12)
(13)
Substituting Eq. (11) and Eq. (13) into Eq. (8) yields
(14)
Dividing both sides of Eq. (14) by gives
(15)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
31 Partial Differential Equations (PDEs)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(16)
The selection of in Eq. (16) is essential to obtain nontrivial solutions. The result (16) gives
two distinct ordinary differential equations given by
(17)
(18)
To determine the function , we solve the second order linear ODE
(19)
The auxiliary of Eq. (12) is
There the solutions of Eq. (17) is
(20)
Similarly we have
(21)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(22)
Similarly the condition gives
(23)
Now we solve Eq. (20) along with the conditions given in Eq. (22) and Eq. (23)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (20) is
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
32 Partial Differential Equations (PDEs)
Consequently, we have Eq. (21) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(24)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (24)
with respect to is
(25)
To determine , we substitute in Eq. (24) and by using the initial condition
we obtain
except
To determine , we substitute in Eq. (25) and by using the initial condition
we obtain
Substituting the values of s and s into Eq. (17), we have
(26)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
33 Partial Differential Equations (PDEs)
Substituting Eq. (26) into Eq. (4) we get
This is our required solution.
Verification:
Since Eq. (26) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (26) twice w.r.t and twice w.r.t we get
(27)
(28)
From Eq. (27) and Eq. (28) we observe that
Now for BCs substitute in Eq. (26) we have
for the Eq. (26), we have
Now for ICs substitute in Eq. (26) we have
putting into , we have
This implies that Eq. (26) is the solution of Eq. (1-3).
Problem 2: Consider the One dimensional Wave equation subject to Dirichlet boundary
conditions as
(1)
BCs (2)
ICs (3)
To convert the inhomogeneous BCs into homogenous BCs we use the following transformation
here we notice that
The above transformation reduces to
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
34 Partial Differential Equations (PDEs)
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
(6)
Now differentiate twice Eq. (4) w.r.t we get
(7)
Eq. (6) and Eq. (7) implies that
For BCs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (4) at we get
For ICs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
(8)
BCs (9)
ICs (10)
According to Method of Separation of Variables (MSV) we assume the following trial solution
(11)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
35 Partial Differential Equations (PDEs)
Differentiating both sides of Eq. (11) twice with respect to t and twice with respect to x we
obtain
(12)
(13)
Substituting Eq. (11) and Eq. (13) into Eq. (8) yields
(14)
Dividing both sides of Eq. (14) by gives
(15)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(16)
The selection of in Eq. (16) is essential to obtain nontrivial solutions. The result (16) gives
two distinct ordinary differential equations given by
(17)
(18)
To determine the function , we solve the second order linear ODE
(19)
The auxiliary of Eq. (12) is
There the solutions of Eq. (17) is
(20)
Similarly we have
(21)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(22)
Similarly the condition gives
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
36 Partial Differential Equations (PDEs)
(23)
Now we solve Eq. (20) along with the conditions given in Eq. (22) and Eq. (23)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (20) is
Consequently, we have Eq. (21) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(24)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (24)
with respect to is
(25)
To determine , we substitute in Eq. (24) and by using the initial condition
we obtain
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
37 Partial Differential Equations (PDEs)
To determine , we substitute in Eq. (25) and by using the initial condition
we obtain
except
Substituting the values of s and s into Eq. (17), we have
Substituting the above expression into Eq. (4) we get
(26)
This is our required solution.
Verification:
Since Eq. (26) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (26) twice w.r.t and twice w.r.t we get
(27)
(28)
From Eq. (27) and Eq. (28) we observe that
Now for BCs substitute in Eq. (26) we have
for the Eq. (26), we have
Now for ICs substitute in Eq. (26) we have
putting into , we have
This implies that Eq. (26) is the solution of Eq. (1-3).
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
38 Partial Differential Equations (PDEs)
1.6. One-dimensional Wave Equation with Inhomogeneous Neumann BCs: In
this section we will consider the case where the boundary conditions of the vibrating string are
inhomogeneous. It is well known that the Method of Separation of Variables requires that the
equation and the boundary conditions are linear and homogeneous. Therefore, transformation
formulas should be used to convert the inhomogeneous boundary conditions to homogeneous
boundary conditions.
In this section we will discuss wave equations where Neumann boundary conditions are not
homogeneous.
In this first type of boundary conditions, the displacements and of a
vibrating string of length are given. We begin our analysis by considering the initial-boundary
value problem
(1)
BCs (2)
ICs (3)
To convert the inhomogeneous boundary conditions of the Eq. (2) to homogeneous boundary
conditions, we simply use the the following transformation formula
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
(6)
Differential Eq. (4) w.r.t. we get
(7)
Differential Eq. (7) w.r.t. we get
(8)
Eq. (6) and Eq. (8) implies that
For BCs, evaluate Eq. (7) at we obtained
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
39 Partial Differential Equations (PDEs)
similarly evaluate Eq. (7) at we get
For ICs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
BCs
ICs
In view of the above system, the method of separation of variables can be easily used in the
above system as discussed before. Having determined of the above system, the wave
function of Eq. (1-3) follows immediately upon substituting into Eq. (4).
Now, we consider two types of Wave equation subject to different type of initial conditions, the
general form of these two types of problems given below:
Type 1:
BCs
ICs
Type 2:
BCs
ICs
Note: Must use transformation in the end to convert the solution into
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
40 Partial Differential Equations (PDEs)
Problem 1: Consider the One dimensional Wave equation subject to Neumann boundary
conditions as
(1)
BCs (2)
ICs (3)
To convert the inhomogeneous BCs into homogenous BCs we use the following transformation
here we notice that
The above transformation reduces to
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
(6)
Differential Eq. (4) w.r.t. we get
(7)
Differential Eq. (7) w.r.t. we get
(8)
Eq. (6) and Eq. (8) implies that
For BCs, evaluate Eq. (7) at we obtained
similarly evaluate Eq. (7) at we get
For ICs, evaluate Eq. (4) at we obtained
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
41 Partial Differential Equations (PDEs)
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
(9)
BCs (10)
ICs
According to Method of Separation of Variables (MSV) we assume the following trial solution
(11)
Differentiating both sides of Eq. (11) twice with respect to t and twice with respect to x we
obtain
(12)
(13)
Substituting Eq. (11) and Eq. (13) into Eq. (8) yields
(14)
Dividing both sides of Eq. (14) by gives
(15)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(16)
The selection of in Eq. (16) is essential to obtain nontrivial solutions. The result (16) gives
two distinct ordinary differential equations given by
(17)
(18)
To determine the function , we solve the second order linear ODE
(19)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
42 Partial Differential Equations (PDEs)
The auxiliary of Eq. (19) is
There the solutions of Eq. (17) is
(20)
Similarly we have
(21)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(22)
Similarly the condition gives
(23)
Now we solve Eq. (20) along with the conditions given in Eq. (22) and Eq. (23), first we
differentiate Eq. (20), w.r.t we get
(24)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (20) is
Consequently, we have Eq. (21) is
Now, combining the results and we obtain the infinite sequence of product
functions
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
43 Partial Differential Equations (PDEs)
Using principle of super position the general solution by
(25)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (25)
with respect to is
(26)
To determine , we substitute in Eq. (25) and by using the initial condition
we obtain
and except
To determine , we substitute in Eq. (26) and by using the initial condition
we obtained
Substituting the values of s and s into Eq. (25), we have
Substituting the above expression into Eq. (4) we get
(27)
This is our required solution.
Verification:
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
44 Partial Differential Equations (PDEs)
Since Eq. (27) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (27) twice w.r.t and twice w.r.t we get
(28)
(29)
From Eq. (28) and Eq. (29) we observe that
Now for BCs substitute in we have
for the expression , we have
Now for ICs substitute in Eq. (27) we have
putting into , we have
This implies that Eq. (27) is the solution of Eq. (1-3).
Problem 2: Consider the One dimensional Wave equation subject to Neumann boundary
conditions as
(1)
BCs (2)
ICs (3)
To convert the inhomogeneous BCs into homogenous BCs we use the following transformation
here we notice that
The above transformation reduces to
(4)
Differential Eq. (4) w.r.t. we get
(5)
Differential Eq. (5) w.r.t. we get
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
45 Partial Differential Equations (PDEs)
(6)
Differential Eq. (4) w.r.t. we get
(7)
Differential Eq. (7) w.r.t. we get
(8)
Eq. (6) and Eq. (8) implies that
For BCs, evaluate Eq. (7) at we obtained
similarly evaluate Eq. (7) at we get
For ICs, evaluate Eq. (4) at we obtained
similarly evaluate Eq. (5) at we get
The transformed initial boundary value problem (IBVP) is
(9)
BCs (10)
ICs
According to Method of Separation of Variables (MSV) we assume the following trial solution
(11)
Differentiating both sides of Eq. (11) twice with respect to t and twice with respect to x we
obtain
(12)
(13)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
46 Partial Differential Equations (PDEs)
Substituting Eq. (11) and Eq. (13) into Eq. (8) yields
(14)
Dividing both sides of Eq. (14) by gives
(15)
This means that the equality holds only if both sides are equal to the same constant. Therefore,
we set
(16)
The selection of in Eq. (16) is essential to obtain nontrivial solutions. The result (16) gives
two distinct ordinary differential equations given by
(17)
(18)
To determine the function , we solve the second order linear ODE
(19)
The auxiliary of Eq. (19) is
There the solutions of Eq. (17) is
(20)
Similarly we have
(21)
where and are constants. To determine the constants and we use the
homogeneous boundary conditions
The condition gives
(22)
Similarly the condition gives
(23)
Now we solve Eq. (20) along with the conditions given in Eq. (22) and Eq. (23), first we
differentiate Eq. (20), w.r.t we get
(24)
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
47 Partial Differential Equations (PDEs)
and
We exclude since it gives the trivial solution Accordingly, we find
Therefore we have Eq. (20) is
Consequently, we have Eq. (21) is
Now, combining the results and we obtain the infinite sequence of product
functions
Using principle of super position the general solution by
(25)
where the arbitrary constants , are as yet undetermined. The derivative of Eq. (25)
with respect to is
(26)
To determine , we substitute in Eq. (25) and by using the initial condition
we obtain
To determine , we substitute in Eq. (26) and by using the initial condition
we obtained
-
BS Mechanical Engineering
Department of Mathematics Proposed by HITCE University Taxila, Pakistan Prof. Dr. Syed Tauseef Mohyud-Din
48 Partial Differential Equations (PDEs)
and except
Substituting the values of s and s into Eq. (25), we have
Substituting the above expression into Eq. (4) we get
(27)
This is our required solution.
Verification:
Since Eq. (27) is the solution of Eq. (1-3) if it satisfies hole system (PDE, BCs and ICs) therefore
differentiate Eq. (27) twice w.r.t and twice w.r.t we get
(28)
(29)
From Eq. (28) and Eq. (29) we observe that
Now for BCs substitute in we have
for the expression , we have
Now for ICs substitute in Eq. (27) we have
putting into , we have
This implies that Eq. (27) is the solution of Eq. (1-3).