Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-baseddifferential quadrature method is presented for solvingone-dimensional homogeneous first-order non-linear parabolicpartial differential equation. First, the given solution domain isdiscretized by using uniform discretization grid point. Next, byusing Taylor series expansion we obtain central finite differencediscretization of the partial differential equation involving withtemporal variable associated with weighted average of partialderivative concerning spatial variable. From this, we obtain thesystem of nonlinear ordinary differential equations and it islinearized by using the quasilinearization method. Then by usingthe polynomial-based differential quadrature method forapproximating derivative involving with spatial variable atspecified grid point, we obtain the system of linear equation. Thenthey obtained linear system equation is solved by using the LUmatrix decomposition method. To validate the applicability of theproposed method, two model examples are considered and solvedat each specific grid point on its solution domain. The stability andconvergent analysis of the present method is worked by supportedthe theoretical and mathematical statements and the accuracy ofthe solution is obtained. The accuracy of the present method hasbeen shown in the sense of root mean square error norm andmaximum absolute error norm and the local behavior of thesolution is captured exactly. Numerical versus exact solutions andbehavior of maximum absolute error between them have beenpresented in terms of graphs and the corresponding root meansquare error norm and maximum absolute error normpresented in tables. The present method approximates the exactsolution very well and it is quite efficient and practically wellsuited for solving the non-linear parabolic equation. Thenumerical result presented in tables and graphs indicates that theapproximate solution is in good agreement with the exactsolution.