Temuer Chaolu

Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach–Adomian–Meyers modified decomposition method

Abstract In this paper we present the generalized Adomian–Rach theorem and the generalized Rach–Adomian–Meyers modified decomposition method for solving multi-order nonlinear fractional ordinary differential equations. We consider different classes of initial value problems for nonlinear fractional ordinary differential equations, including the case of real-valued orders and another case of rational-valued orders, which are solved by the present method. This method can treat any analytic nonlinearity. The coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme, which does not involve integration operations compared with the classic Adomian decomposition method.

Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation

In this paper, we successfully construct the new exact traveling wave solutions of the coupled Schrodinger-Boussinesq equation by using the extended simplest equation method. The exact traveling wave solutions with double arbitrary parameters, are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.

The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations

In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Pade approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Pade approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution.

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

Abstract In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada–Kotera, Sawada–Kotera–Parker–Dye, Caudrey–Dodd–Gibbon, Kaup–Kupershmidt, Kaup–Kupershmidt–Parker–Dye, and the Ito forms.

A new method for solving boundary value problems for partial differential equations

This paper proposes a symmetry-iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.

A method to select the initial guess solution, auxiliary linear operator and set of basic functions of homotopy analysis method

In this paper, we clearly give a method to determine the initial guess solution, auxiliary linear operator and set of basic functions of the homotopy analysis method one by one. As application, we considered the Thomas-Fermi equation and non-linear heat transfer equation to illustrate the effectiveness and convenience of the suggested method.

Exact Interaction Solutions of an Extended (2+1)-Dimensional Shallow Water Wave Equation*

Applying the consistent Riccati expansion method, the extended (2+1)-dimensional shallow water wave equation is proved consistent Riccati solvable and the exact interaction solutions including soliton-cnoidal wave solutions, solitoff-typed solutions are obtained. With the help of the truncated Painleve expansion, the corresponding nonlocal symmetry is also given, and furthermore, the nonlocal symmetry is localized by prolonging the related enlarged system.

Homotopy series solutions of perturbed PDEs via approximate symmetry method

We show that the two couple equations derived by approximate symmetry method and approximate homotopy symmetry method are connected by a transformation for the perturbed PDEs. Consequently, approximate homotopy series solutions can be obtained by acting the transformation on the known solutions by approximate symmetry method. Applications to the Cahn-Hilliard equation illustrate the effectiveness of the transformation.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

Approximate Solitary Wave Solutions for a Perturbed BBM Equation by a Hybrid Approach

Based on the approximate symmetry group method, the differential form Wu’s method and a generalized Riccati equation with variable coefficients expansion method, a hybrid approach for obtaining the approximate solitary wave solutions of a perturbed BBM equations is introduced.