A METHOD BASED ON BERNSTEIN POLYNOMIAL FOR SOLVING NONLINEAR VOLOTERRA-FREDHOLM INTEGRAL EQUATIONS OF SECOND

Abstract

In this thesis, we discussed numerical solutions of nonlinear Volterra-Fredholm integral equations by using Bernstein polynomial collocation method. Nonlinear Volterra-Fredholm integral equations which cannot be easily evaluated analytically. This thesis was concerned with numerical method (Bernstein polynomial collocation method). Bernstein polynomial collocation method were utilized to convert the Nonlinear Volterra-Fredholm integral equation into a system of nonlinear algebraic equations and the resulting nonlinear algebraic equations were solved by using newton iteration technique to compute the Bernstein coefficients. The presented concept and method was verified by different examples, where theoretical results are numerically confirmed. The numerical results of six test problems, for which the exact solutions are known, are considered to verify the accuracy and the efficiency of the proposed method. The numerical results were compared with the exact solutions and with other method used for solving nonlinear VFIE based on absolute error. Finally the numerical results were demonstrated by table, in order to show the reliability of proposed method. From the results of the study as the values of the degree n, increases and the error were decreasing and computational costs were increases to obtain more accurate results. Lastly from the result, we have seen that BPCM is converge to exact solution as number of degree is increased. This thesis can be extended to solve nonlinear volterra-fredholm integro differential equation and also can be extended for the numerical solutions of two dimensional nonlinear volterra-fredholm equations using these methods.