Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

Abstract

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

“…We present the comparison between that method and the present one in Figure 1 and Table 2 to demonstrate higher accuracy obtained by our method as compared to that reported by Merdan et al [36] while computing the solution of the fractional-order van der Pol oscillator model. Our quantitative Comparison of maximum errors in the obtained results for M = 8, k = 1 and r = 7 iterations with published results [37,39,41,44,45,[49][50][51][52] analysis presented in Table 1 provides an evidence that the linearized shifted Gegenbauer wavelets method is more accurate and efficient than the multistep differential transform method since it shows lesser absolute error for 1 = 0.1. Figure 1a shows that our approximate solution converges uniformly and approaches that computed by Runge Kutta method of order four (RK −4 ) with the derivative order → 2.…”

“…We also illustrate the trends of the absolute error for k = 1 and M = 10 in Figure 1d where we witness the growth of error with an increasing time since it relates to an initial value problem. Comparison in the absolute error for M = 8, k = 1 and r = 7 iterations with published results [37,40] The generalized Burger's-Huxley model relates to a significant phenomenon that governs the interaction among reaction mechanisms, diffusion transport, and convection effects. The model consists a reaction diffusion term as given on the right hand side of Equation (37).…”

“…Comparison of maximum errors in the obtained results for M = 8, k = 1 and r = 7 iterations with the results in [37,40,48] Abbreviation: LSGWM, linearized shifted Gegenbauer wavelets method.…”

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

“…The accuracy of the current scheme is verified by the error norms L ∞ , L 2 and experimental order of convergence (EOC) [22,23]:…”

Exponential B-spline collocation method for solving the generalized Newell-Whitehead-Segel equation

“…If U(x, t) represents the true solution and u(x, t) is the approximate solution, then using exponential B-spline functions we let [10,19,20]…”