Solving a Class of Singularly Perturbed Partial Differential Equation by Using the Perturbation Method and Reproducing Kernel Method

Abstract

We give the analytical solution and the series expansion solution of a class of singularly perturbed partial differential equation (SPPDE) by combining traditional perturbation method (PM) and reproducing kernel method (RKM). The numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

A Class of Block Multi-derivative Numerical Techniques for Singular Advection Equations

“…2 The superiority of the proposed methods could be established in the table above. From the table above, it could also be From table 6 above, it could also be observed that the Method shows some superiority in-terms of accuracy when compared with the method of Wang et al [11] and also compares favourably 3 (x, t) is the coordinate of x and t, u(x, t) is the exact solution, UM is the Upwind method, LM is the Lax method, Method represents the solution using the Method and Relative error=| u(x, t) -Method |/u(x, t). Percentage error = Relative error × 100%.…”

“…In this section, the standards methods of Upwind, Lax method, the method of Wang et al [11] and the Method were employed successfully for solving the advection equations. The numerical results of the proposed method show some superiority over other methods in-terms of accuracy and reliability.…”

“…Moreover, the Method is also effective for solving the equation with singular terms and other nonlinear singular perturbation problems. Wang et al [11] used a perturbation method and reproducing kernel method in solving a class of singularly perturbed partial differential equation. From Table 3 above, it could be observed that the Method shows high computational strength as it compares favourably well with some known standard methods reported.…”

“…This work contains a new class of multi-derivative method. The multi-derivative method was considered up to the second Table 6: Numerical Comparison for Example 4 for v(x, t) = u(x, t), ∆x = 0.01, ∆t = 0.01, = 10 −4 (x, t) u(x, t) UM LM Wang et al [11] Method (40) Absolute error (0.0000,0.0000) 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 (0.0100,0.0100) 0.01000100000 0.01009999010 0.01009999010 0.01000000000 0.01000010000 9.000000 E -07 (0.0300,0.0300) 0.03000300000 0.03089993849 0.03089993849 0.03000000000 0.03000030000 2.700000 E -06 (0.0500,0.0500) 0.05000500000 0.05249984036 0.08002383910 0.05000000000 0.05000050000 4.500000 E -06 (0.0600,0.0600) 0.06000600000 0.06359977227 0.1648091770 0.06000000000 0.06000060000 5.400000 E -06 (0.0800,0.0800) 0.08000800000 0.08639959684 0.4698987551 0.08000000000 0.08000080000 7.200000 E -06 (0.1000,0.1000) 0.1000100000 1.009904485 0.8868283796 0.1000000000 0.1000010000 4.500000 E -06 CPU time NA 0.400000s 0.171875s NA 0.078125s derivative (k = 2). The method was developed using some selected off-step points in its formation (this is to improve its accuracy), evaluation such as collocation and interpolation of these points were carried out and a system of algebraic equations was obtained for each value of k together with number of off-step points considered.…”

The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning Approach

“…e area of SP is a field of increasing interest to applied mathematicians and computer mathematics scientists. During the last decades, considerable contributions on the topic and its applications have been made by several researchers [1,[32][33][34][35][36][37][38][39][40][41]. Note that considerable research work in special topics of SP nonlinear differential equations has been recently presented [16][17][18][19][20][21].…”