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The present study investigates an algorithm numerically for finding the solution of partial differential equation with differences involved singular perturbation parameter(SPPDE) on non-uniform grid. Taylor series expansion provides a close approximation of the delay and advance terms in the convection-diffusion terms. After the approximations in shift containing terms, we applied the Crank-Nicolson application on uniform grid in the vertical direction. Subsequently, the resultant system is employed by the method of tension spline on a piece-wise uniform grid. Empirical evidence has shown that the suggested approach exhibits second-order characteristics in both the spatial and temporal dimensions. The effectiveness of derived scheme demonstrated through the solution of examples and the results are compared with existed methods. In the conclusion section, we will discuss the effect of shift parameters behavior for various singular perturbation parameter. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical algorithm dealing with multi‐term time‐fractional Burgers‐type equation involving the Caputo derivative. The proposed method consists of temporal discretization of L2$$ L2 $$ formula and spatial discretization using the exponential B‐splines. The semi implicit approach is applied to discretize the nonlinear term u∂xu$$ u{\partial}_{\mathtt{x}}u $$. We adopt the Von–Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]