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In this paper, the exact solutions of a reduced extended $$(3+1)$$ -dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.

As we known, reproducing kernel method (RKM) has been presented for solving differential equations for initial and boundary value problems. However, the direct application of the RKM presented in the previous works cannot produce good numerical results for Burgers’ equation. To solve this problem, this paper give a modified reproducing kernel method by piecewise technique. The exact solution is given by reproducing kernel functions in a series expansion form, the approximation solution is expressed by n-term summation of reproducing kernel functions. The three numerical experiments results show that the piecewise method is more easily implemented and effective. Some numerical results are also compared with the results obtained by other methods.

This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n-term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective.

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations (PDEs).

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method.

A complete classical symmetry classification and a nonclassical symmetry classification of a class of nonlinear wave equations are given with three arbitrary parameter functions. The obtained results show that such nonlinear wave equations admit richer classical and nonclassical symmetries, leading to the conservation laws and the reduction of the wave equations. Some exact solutions of the considered wave equations for particular cases are derived.

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.

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.