In this paper, a new operational matrix method based on the Taylor polynomials is presented to solve generalized pantograph type delay differential equations. The method is based on operational matrices of integration and product for Taylor polynomials. These matrices are obtained by using the best approximation of function by the Taylor polynomials. The advantage of the method is that the method does not require collocation points. By using the proposed method, the generalized pantograph equation problem is reduced to a system of linear algebraic equations. The solving of this system gives the coefficients of our solution. Numerical examples are given to demonstrate the accuracy of the technique and the numerical results show that the error of the present method is superior to that of other methods.The residual correction technique is used to improve the accuracy of the approximate solution and estimation of absolute error. The estimation aspect of the residual method is useful when the exact solution of the considered equation is unknown and this feature of the method is shown in numerical examples.

The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm-Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational matrices of Taylor polynomials in order to solve linear FVIDEs. By using the operational matrices of integration and product for the Taylor polynomials, the problem for linear FVIDEs is transformed into a system of linear algebraic equations. The solution of the problem is obtained from this linear system after the incorporation of initial conditions. Numerical examples are presented to show the applicability and the efficiency of the method. Wherever possible, the results of our method are compared with those yielded by some other methods.

In this paper, the human immunodeficiency virus (HIV) infection model of CD[Formula: see text][Formula: see text]T-cells is considered. In order to numerically solve the model problem, an operational method is proposed. For that purpose, we construct the operational matrix of integration for bases of Taylor polynomials. Then, by using this matrix operation and approximation by polynomials, the HIV infection problem is transformed into a system of algebraic equations, whose roots are used to determine the approximate solutions. An important feature of the method is that it does not require collocation points. In addition, an error estimation technique is presented. We apply the present method to two numerical examples and compare our results with other methods.

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.

Abstract:In this paper, the differential transform method is extended by providing a new theorem to two-dimensional Volterra integral equations with proportional delays. The method is useful for both linear and nonlinear equations. If solutions of governing equations can be expanded for Taylor series, then the method gives opportunity determine coefficients Taylor series, i.e. the exact solutions are obtained in series form. In illustrate examples the method applying to a few type equations.