A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials

Abstract

In this paper, a new collocation method based on the Pell–Lucas polynomials is presented to solve the parabolic-type partial Volterra integro-differential equations. According to the method, it is assumed that the solution of this equation is in the formu2N(x,t)≅∑n=0N∑s=0Nan,sQn,s(x,t),Qn,s(x,t)=Qn(x)Qs(t)which depends on the Pell–Lucas polynomials. Next, the matrix representation of the solution is written. Using this matrix form, the matrix representations of the partial derivatives, the matrix representations of the Volterra integral part and the matrix forms of the conditions are also constituted. All obtained matrix forms are substituted in the equation and its conditions. Using equally spaced collocation points in matrix forms of this equation and initial conditions, the equation is reduced to a system of algebraic equations. The solution of this system gives the coefficients of the assumed solution. Additionally, the error analysis for the method is presented. According to this, an upper bound of the errors is determined. Also, the error estimation is made with the help of the residual function. Moreover, the residual improvement technique is also applied. Then, all these procedures are then supported with the examples. The results obtained from these examples are clearly tabulated and graphed. An important aspect of this study is to compare the obtained results with the present method with other results in the literature.