The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.
This study is aimed to develop a new matrix method, which is used an alternative numerical method to the other method for the high-order linear Fredholm integro-differential-difference equation with variable coefficients. This matrix method is based on orthogonal Jacobi polynomials and using collocation points. The improved Jacobi polynomial solution is obtained by summing up the basic Jacobi polynomial solution and the error estimation function. By comparing the results, it is shown that the improved Jacobi polynomial solution gives better results than the direct Jacobi polynomial solution, and also, than some other known methods. The advantage of this method is that Jacobi polynomials comprise all of the Legendre, Chebyshev, and Gegenbauer polynomials and, therefore, is the comprehensive polynomial solution technique
An advanced matrix method is formulated for the solution of Volterra integro-differential equations with weakly singular kernel by using orthogonal Jacobi polynomials. Employing this practical method, it is possible to solve various types of these equations routinely in a systematic fashion. An error estimation procedure is prescribed to estimate the error of the basic Jacobi method and then the error correction term is added to the basic method to obtain more accurate results. Four test experiments are provided to confirm the validity and systematic approach of the advanced method. These experiments also certified that this advanced method surpasses the basic Jacobi method, as well as several alternative approaches.
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