Developing a finite difference hybrid method for solving second order initial-value problems for the Volterra type integro-differential equations

Abstract

It is well known that the study of many processes of the natural sciences can be reduced to solving Volterra integrodifferential equations. Recent studies on certain problems of the environment such as the HIV virus, bird flu virus, and diseases associated with mutations of viruses have become relevant. A solution to such problems is associated with finding solutions of VIDEs. There are several classes of methods for solving IDEs. In contrast to the known methods, this paper developed the finite difference hybrid method by a combination of power series and the shifted Legendre polynomial through a block method which is self-starting and helped in eliminating the problem inherent with finding special predictors to estimate 𝑦 ′ in the integrators. The method was analyzed and the result revealed that the method is consistent, zero stable and convergent. Some test examples were considered and the results compared favorably with some existing methods.

Keywords

• volterra integro differential equations
• finite difference method
• hybrid method
• shifted legendre polynomials
• trapezoidal rule