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A moderate improvement of the standard homotopy perturbation method is given and applied to the large deflection problem of a uniformly loaded circular plate with edge simply supported and fastened. Compared with the general homotopy perturbation method, there are two innovations in our method. On the one hand, the homotopy equations are constructed based on part of equations instead of all equations of the system. On the other hand, a parameter in the system is expanded to finite terms which have more freedom to be selected and it can be used to adjust the accuracy of the approximate solutions. More accurate solutions for the large deflection problem are obtained by the proposed method. The error comparisons with classical perturbation solutions show the efficiency and convenience of the proposed homotopy perturbation method.

Purpose – The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field. Design/methodology/approach – The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials. Findings – The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems. Originality/value – In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

The integrability and multi-shock wave solutions of the DJKM equation are studied by means of Bell polynomials scheme, Hirota bilinear method, and symbolic computation. A more generalized bilinear system of the DJKM equation is constructed via Bell polynomials scheme. Moreover, Lax pair and infinite conservation laws of this equation are first obtained via its corresponding Bell-polynomials-type Backlund transformation. Furthermore, the multi-shock wave solutions are also obtained by applying standard Hirota bilinear method, and the propagation and collision of shock waves are graphically demonstrated by graphs.

In the present paper, a direct and unifying Bell polynomial approach is employed to find the bilinear representations, bilinear Backlund transformation and Lax pair of a generalised variable-coefficient fifth-order Korteweg–de Vries equation. Note that the integrable constraint conditions on the variable coefficients can be naturally found in the procedure of applying the Bell polynomials. In addition, with the help of the Hirota bilinear method, the N -soliton solutions of this equation are also obtained.

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.

In this paper, the problem of viscous heating in plane Couette flow is considered by the homotopy perturbation method. The non-linear terms are expanded to Taylor series of the homotopy parameter. The obtained solutions are shown graphically and are compared with the exact solutions. The obtained results illustrate the efficiency and convenience of the method.

Through symbolic computation with Maple, two classes of lump solutions, rationally localized in all directions in space, are presented for a new extended (2[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq equation. Analyticity of the solutions is naturally achieved, and particularly, taking special choices of the involved parameters will guarantee the positiveness of the constant term in the quadratic function f. Moreover, it deserves a note that one parameter in f plays an important role in order to maintain the positiveness of the quadratic function f. As illustrative examples, two particular lump solutions with specific values of the involved parameters are worked out and their three-dimensional plots, contour plots, x-curves and y-curves are made.

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations (PDEs) with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu’s method, in differential equations.

In this paper, we used a different initial guess solution then others, to obtain very good approximate solution of the Fitzhugh-Nagumo equation by the homotopy analysis method. Thus, we further illustrate the freedom to select the initial guess solution, i.e. illustrate ffectiveness and convenience of homotopy analysis method.