A Comparison Between Analytical and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind with Weakly Singular Kernel by using the Non-polynomial Spline Functions

Abstract

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This thesis will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piecewise polynomials of degree joined together at the break points with) (continuous derivatives. The break points of splines are called knot,spline function can be integrated and differentiated due to being piecewise polynomials and can easily stored and implemented on digital computer , non-polynomial spline function a piecewise is a blend of trigonometric, as well as, polynomial basis function ,which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this thesis is to compare between analytical solution and numerical solution to solve Volterra integral equations of second kind using the fifth order non-polynomial Spline functions and linear Volterra integral equations with weakly singular kernel using the sixth order non-polynomial spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with other numerical methods for analytical solutions. We show finally by comparison of numerical results, simplicity and efficiency of these methods.