Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function ()
1. Introduction
Integro-differential equations (IDEs) have attracted growing interest over the years because of the mathematical application in real life problems. Mathematical modeling of real life problems usually resulted in fractional equations. Many mathematical formulations of physical phenomena contain integro-differential equations. These equations arise in many fields like Physics, Astronomy, Potential theory, Fluid dynamics, Biological models and Chemical kinetics. Integro-differential equations contain both integral and differential operators. The derivatives of the unknown functions may appear to any order (see [1] and [2] ). [3] obtained solution of an integro-differential equation arising in oscillating magnetic field using Homotopy perturbation method. Galerkin method is a powerful tool for solving many kinds of equations in various fields of science and engineering. It is one of the most important weighted residual methods inverted by Russians mathematicians Boris Grigoryevrich Galerkin. Recently, various Galerkin algoriyhm have been applied in numerical solution of integral and integro-differential equations. The following methods that are based on the Galerkin ideas, includes Galerkin Finite Element [4] , iterative Galerkin with hybrid functions [5] , Crank-Nicolson least squares Galerkin [6] , and Legendre Galerkin [7] . [8] published a note on three numerical procedures to solve Volterra integro-differential equations on structural analysis.
2. Problem Considered
We consider the higher order linear integro-differential equation as follows:
(1)
Subject to the following conditions
(2)
where
are constant coefficients,
and
are lower and upper limits of integration,
is a constant parameter and
is a function of two variables x and t called the kernel,
is a known function and
is the unknown function to be determined.
3. Definitions
Integro-differential equation
An integro-differential equation is an equation involving both integral and derivatives of a function. Example of such equation is stated below:
(3)
Galerkin method
Galerkin method is a method of determining coefficient
in a power series solution of the form:
(4)
of the ordinary differential equation
so that
, the result of applying the ordinary differential operator to
, is orthogonal to every
for
Chebyshev Polynomial
The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted by
. The Chebyshev polynomial of the first kind denoted by
is defined by the contour integral
Where the contour encloses the origin and is traversed in a counter clockwise direction.
Orthogonal over a set
A set of function
is said to be orthogonal over a set of points
with respect to the weight function
, if
Orthogonal over an interval
A set of functions
is said to orthogonal on an interval
with respect to the weight function
, if
Approximate solution
Approximate solution is used for the expression obtained after the unknown constants have been generated and substituting back into the assumed solution. It is hereby call approximate solution since it is a reasonable approximation to the exact solution.
4. Construction of Orthogonal Polynomials
In this section, we constructed orthogonal polynomials
, valid on the interval
with the leading term
Then, starting with
, (5)
Thus, we find the linear polynomials
, with leading term x, is written as
, (6)
where,
is a constant to be determined. Since
and
are orthogonal, we have,
using (5) and (6).
From the above, we have,
Hence, (6) gives,
Now, the polynomials
, of degree 2 and the leading term
is written as
(7)
where the constants
and
are determined by using orthogonality conditions
(8)
Since
is orthogonal to
, we have
(9)
Since,
The above equation gives
(10)
Again, since
is orthogonal to
, we have
Thus, using (7), we obtain
(11)
Since
and
are known, (7) determines
. Proceeding in the same way, the method is generalized and we have,
(12)
where the constants
and so chosen that
is orthogonal to
These conditions yield,
(13)
Few terms of orthogonal polynomials valid in the interval [−1, 1] are given below.
etc.
5. Demonstration of Orthogonal Galerkin Method on General Problem Considered
In this section, we considered (1) and (2).
Here we assumed an approximate solution of the form
(14)
where
are the orthogonal polynomial constructed and valid in the interval [−1, 1].
Thus, differentiating (14)/with respect to x, n times, we have
(15)
Substituting (14) and (15) into (1), we obtain
(16)
We determined the unknown coefficients
using the Galerkin idea by multiplying both sides of (16) by
and then integrating with respect to x from −1 to 1.
Thus, we obtain
(17)
This process generates a system of linear equations for the unknown
together with the conditions
(18)
for the same number of equations in the linear system.
The unknown parameters are determined by solving the system (17) and (18). The values of the constants obtained are then substituted back into (14) to get the required approximate solution for the appropriate order.
6. Numerical Experiments
In this section, we consider four selected problems for experimenting and compare our results with existing results.
Numerical example 1
We consider the Volterra integro-differential equations of the second kind of the form:
(19)
together with the condition given as
(20)
The exact solution is given as
Here we solved example 1 for case
.
Thus, Equation (14) becomes
(21)
Substituting the values of
, we obtain
(22)
and,
(23)
Substituting (23) into (19) for case N = 4, we obtain
(24)
Thus, evaluating the integral in (24) and simplifying, we obtain
(25)
The unknown coefficients
are determined using the Galerkin idea by multiplying both sides of (25) by
and then integrating the resulted equation between x = −1 to x = 1.
For case j = 1, we multiplied both sides of (25) by (2x − 1) and then integrating the resulted equation between x = −1 to x = 1, to obtain
(26)
For case j = 2, we multiplied both sides of (25) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(27)
For case j = 3, we multiplied both sides of (25) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(28)
For case j = 4, we multiplied both sides of (25) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(29)
Now, using the condition given in (22), we obtain
(30)
Hence, (26)-(30) are then solved to obtain the unknown constants
which are then substituted to the approximate Equation (22).
Again, we solved (1) and (2) for case N = 6 by re-writing (21) as:
(31)
Hence, (31) becomes
(32)
And,
(33)
Thus substituting (32) and (33) into (19), we obtain
(34)
Thus, evaluating the integral in (34) and simplifying, we obtain
(35)
The unknown coefficients
are determined using the Galerkin idea by multiplying both sides of (35) by
and then integrating the resulted equation between x = −1 to x = 1.
For case j = 1, we multiplied both sides of (35) by (2x − 1) and then integrating the resulted equation between x = −1 to x = 1, to obtain
(36)
For case j = 2, we multiplied both sides of (35) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(37)
For case j = 3, we multiplied both sides of (35) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(38)
For case j = 4, we multiplied both sides of (35) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(39)
For case j = 5, we multiplied both sides of (25) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(40)
For case j = 6, we multiplied both sides of (35) by
and then integrating the resulted equation between x = −1 to x = 1, to obtain
(41)
Now, using the condition given in (22), we obtain
(42)
Hence, (36)-(42) are then solved to obtain the unknown constants
which are then substituted to the approximate equation (32). More values of N are computed follow the same procedure and the results obtained are tabulated below.
Example 2:
With the conditions
and
, The exact solution is
.
Example 3: Consider the Fredholm integro-differential equation (See [2] )
Together with the conditions
;
;
. The exact solution is
.
· Denotes the results are not available for comparison
· Denotes Results are not available for comparison
Example 4: Consider the Fredholm integro-differential equation (See [2] )
With the following conditions
;
;
;
. The exact solution is
.
· Denotes Results are not available for comparison
· Denotes Results are not available for comparison
7. Discussion of Results
The approximate solution obtained by means of Galerkin method is a finite power series which can be in turn expressed in closed form of exact solution as the degree of the approximant increases. The finite series solution is obtained for each problem considered by increasing the value of N, which in turn converges to closed form of exact solution, the absolute errors obtained tend to zero and ensures stability of our method (See Tables 1-8). Also, from the results obtained by [2] , our method proved superior to [2] . As N increases, the results obtained in some cases converged. It proves a very efficient method for the problems attempted, for which the form of the solution is known.
Table 1. Numerical results and absolute errors of example 1 for case N = 4.
Table 2. Numerical results and absolute errors of example 1 for case N = 6.
Table 3. Numerical results and absolute errors of example 2 for case N = 4.
Table 4. Numerical results and absolute errors of example 2 for case N = 4.
Table 5. Numerical results and absolute errors of example 3 for case N = 4.
*Denotes the results are not available for comparison.
Table 6. Numerical results and absolute errors of example 3 for case N = 10.
*Denotes the results are not available for comparison.
Table 7. Numerical results and absolute errors of example 4 for case N = 4.
*Denotes the results are not available for comparison.
Table 8. Numerical results and absolute errors of example 4 for case N = 10.
*Denotes the results are not available for comparison.
8. Conclusion
In this work, we have proposed the Galerkin method for solving both the boundary and initial value problems for a class of higher order linear and nonlinear Volterra and Fredholm integro-differential based on the constructed orthogonal polynomials as basis function. Illustrative examples are included to demonstrate the validity and applicability of the technique and the tables of results presented reveal that the absolute error decreases when the degree of approximation increases. Furthermore, since basis functions constructed are polynomials, the values of the integrals for the nonlinear integro differential equations are calculated as approximately close to the exact solutions.