Multistep block methods for solving volterra integro-differential equations of second kind

Numerical solutions of Volterra integro-differential equations (VIDEs) by using multistep block methods are proposed in this thesis. The two point one-step block method, two point two-step block method and two point three-step block method are derived by using the Lagrange interpolating polynomial. The generated multistep block methods will estimate the solution of VIDEs at two points simultaneously in a block by using constant step sizes. The source code for solving VIDEs are developed by using C programming. In VIDEs the unknown functions appear under the differential and integral sign, so the combinations of multistep block methods with numerical quadrature rules are applied. The multistep block methods are used to solve the ordinary differential equation (ODE) part and quadrature rules are applied to calculate the integral part of VIDEs. The method developed has solved for linear and nonlinear second kind VIDEs. The type of numerical quadrature rules used for solving the integral part of VIDEs is of Newton-Cotes type. Thus, the quadrature rules of suitable order are used to be paired with the multistep block methods. Two different approaches are proposed to solve for two cases where kernel equal or not equal one. The stability region of the combination methods are studied. Numerical problems are presented to show the performance of the proposed method. The results indicated that the proposed method is suitable for solving both linear and nonlinear VIDEs.

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