A Taylor operation method for solutions of generalized pantograph type delay differential equations

Abstract

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.

Spectral Tau method for solving general fractional order differential equations with linear functional argument

“…Equations (24) and (27) generate (m) and (N − m + 1) set of algebraic equations, respectively. Consequently, the unknown coefficients of the vector A in (16) can be calculated.…”

“…The general form of argument (mixed type equations) have been reported by several mathematicians, where Grbz et al used Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments [26]. Yuzbasi has reported a solution of the generalized pantograph type delay differential equations with linear functional arguments [27]. Reutskiy used the backward substitution method for multi-point problems with linear Volterra-Fredholm integro-differential equations of the neutral type [28].…”

A Galerkin-like method for solving linear functional differential equations underinitial conditions

“…Bernoulli polynomials [14], a Taylor polynomial approach [15], a collocation method involving Lucas polynomials [16], variational iteration method [17], and an operational matrix method using polynomials in the standard basis [18]. In addition to these, pantograph equations have been solved using shifted orthonormal Bernstein polynomials [19].…”

A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

“…However, the rapid development of the theory and applications of those equations did not come until after the Second World War and continues till today. Additionally, the numerical solution of delay and advanced differential equations of fractional order have been reported by many researchers [7][8][9][10][11][12][13][14][15]. Differential equations of advanced argument had fewer contributions in mathematics research, compared to delay differential equations, which had a great development in the last decade [16][17][18][19][20].…”