Abstract
In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.
Highlights
Most physical phenomena arising in various fields of engineering and science are modeled by nonlinear partial differential equations (NPDEs)
Further to prove the accuracy and reliability of the proposed scheme comparisons of the numerical results are made with the exact solutions and some traditional methods, including optimal homotopy asymptotic method (OHAM) [10], adomian decomposition method (ADM) [11], homotopy perturbation method (HPM) [12], and collocation based radial base functions method (CBRBF) [13]
We consider the generalized Burgers0-Fisher equation transformed into nonlinear ordinary differential equation (NODE) given by Equation (13) with the initial condition given by (2)
Summary
Most physical phenomena arising in various fields of engineering and science are modeled by nonlinear partial differential equations (NPDEs). The investigation of solutions to NPDEs has attracted much attention due to their potential applications and many numerical schemes have been proposed, see for example [1,2,3,4]. The generalized Burgers0-Fisher equation is one of the important NPDE which appears in various applications, such as fluid dynamics, shock wave formation, turbulence, heat conduction, traffic flow, gas dynamics, sound waves in viscous medium, and some other fields of applied science [5,6,7,8,9,10]. The generalized Burgers0-Fisher equation is of the form [10,11,12].
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