Abstract
The homotopy analysis method (HAM) is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained. The efficiency and accuracy of the method are verified by using two famous examples: coupled Burgers and mKdV equations. The obtained results show that the homotopy perturbation method is a special case of homotopy analysis method.
Highlights
Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering
The nonlinear oscillation of earthquake can be modeled with fractional derivatives, and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow
The differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena 1, 2. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, and on the history of the previous time which can be successfully achieved by using fractional calculus
Summary
Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering. In 1992, Liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, homotopy analysis method 6– This method has been successfully applied to solve many types of nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids , the KdV-type equations , finance problems , fractional Lorenz system , and delay differential equation. Numerical methods use discretization which gives rise to rounding off errors causing loss of accuracy and requires large computer memory and time This computational method yields analytical solutions and has certain advantages over standard numerical methods. We extend the application of HAM to discuss the explicit numerical solutions of a type of nonlinear-coupled equations with parameters derivative in this form:.
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