Polynomial And Quadrature Method For Solving Linear Integraland Integro-Differential Equations
This thesis considers linear Volterra-Fredholm integral equations (lEs) and linear Volterra-Fredholm integro-differential equation (IDEs) of order one, two and high order, with boundary condition. Polynomial approximation method together with Gauss- Legendre quadrature formula (QF) are proposed t...
Saved in:
| Main Author: | |
|---|---|
| Format: | Thesis |
| Language: | English |
| Subjects: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | This thesis considers linear Volterra-Fredholm integral equations (lEs) and linear
Volterra-Fredholm integro-differential equation (IDEs) of order one, two and high order,
with boundary condition. Polynomial approximation method together with Gauss-
Legendre quadrature formula (QF) are proposed to find the approximate solution of those
equations. The Legendre truncated series of order n is chosen as the basis function to
estimate the solution. Gauss-Legendre QF and the collocation method are applied to
reduce the Volterra-Fredholm lEs into the system of linear algebraic equation. The later
equation is then solved by Gauss elimination method.
For the first and second order Volterra-Fredholm IDEs, reduction method is used
before applying the polynomial approximation method with quadrature forniula. A
different approach is used for the high order Volterra-Fredholm IDEs. The operational
matrix of derivative for Legendre polynomials is used to transform the high order
Volterra-Fredholm IDEs into matrix equation. Gauss-Legendre QF and the collocation
method are used to form a system of linear algebraic equation. The system is then solved
by Gauss elimination method. For the first and second order Volterra-Fredholm IDEs,
convergence analysis and rate of convergence of the proposed method are obtained in the
class of smooth functions C' [u, h] with the Chebyshev none.
Test problems are provided to show the accuracy and validity of the proposed
method. The results show that the error of the proposed method decreased faster as n
increases accept for the equation that has 4.5 in its exact solution and the equation that
appear many derivatives of unknown function. Some test problems show that the
proposed method able to solve exactly the problems when the solution is the polynomial
form. The errors for some test problems are compared with errors obtained by other
methods. The operational matrix of derivative for Legendre polynomials solved the
problems without reduce it into integral equation and make fewer jobs. All the numerical
results are obtained using Maple 17. |
|---|