Exact wave solutions for a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation

In this work, we investigate a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation. Based on the simplified Hirota bilinear method, we first construct its soliton solutions. Meanwhile, we correct the formula of N-soliton solution for this equation. On the basis of these solitons we further calculate its lump solutions, periodic waves. Meanwhile, rogue waves as well as interaction solutions of this equation are also obtained by a direct algebraic method. Some figures are given to display the behavior of these solutions.     

Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev–Petviashvili equation

Kadomtsev–Petviashvili equations with variable coefficients can be used to characterize many nonlinear phenomena in fluid dynamics and plasma physics more realistically than the equations with constant coefficients. Hereby, a generalized variable-coefficient Kadomtsev–Petviashvili equation with nonlinearity, dispersion and perturbed terms is investigated. Transformations, of which the consistency conditions are exactly the Painlevé integrability conditions, to the Korteweg–de Vries equation, cylindrical Korteweg–de Vries equation, Kadomtsev–Petviashvili equation and cylindrical Kadomtsev–Petviashvili equation are presented by formal dependent variable transformation assumptions. Using the Hirota bilinear method, from the variable-coefficient bilinear equation, the multi-solitonic solution, auto-Bäcklund transformation and Lax pair for the variable-coefficient Kadomtsev–Petviashvili equation are obtained. Moreover, the influence of inhomogeneity coefficients on solitonic structures and interaction properties is discussed for physical interest and possible applications.     

A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

In this paper, an efficient and accurate meshless method is proposed for solving the time fractional advection–diffusion equation with variable coefficients which is based on the moving least square (MLS) approximation. In the proposed method, firstly the time fractional derivative is approximated by a finite difference scheme of order $O\left({\left(\delta t\right)}^{2?\alpha }\right),0<\alpha \le 1$ and then the MLS approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the Caputo sense. Also, the validity of the proposed method is investigated in error analysis discussion. The main aim is to show that the meshless method based on the MLS shape functions is highly appropriate for solving fractional partial differential equations (FPDEs) with variable coefficients. The efficiency and accuracy of the proposed method are verified by solving several examples.     

Locally distributed control of wave equations with variable coefficients

For wave equations with variable coefficients on regions which are not necessarily smooth, we obtain a sufficient condition for the subregion on which the application of control will yield the exact controllability property by using piecewise multiplier method and Riemannian geometry method. Some examples are presented.     

Spectral collocation solution of a generalized Hirota–Satsuma coupled KdV equation

In this paper the solution of a generalized Hirota–Satsuma Korteweg–de Vires (KdV) equation using a pseudospectral method is presented. To reduce roundoff error we use some preconditionings. Firstly, we discretize the equation in space to obtain a system of time-dependent ordinary differential equations. Secondly, we solve the obtained system of ordinary differential equations using the fourth-order Runge–Kutta method. The method works very well because the absolute errors are very small.     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Runge–Kutta method with equation dependent coefficients

The family of the simplest three-stage explicit Runge–Kutta methods is examined by a conveniently adapted form of the exponential fitting approach. We obtain versions whose unusual feature is that their coefficients are no longer constant, as in the standard version, but depend on the equation to be solved. Two mathematical properties of the new versions are specially helpful for applications. Firstly, although in general the order is three, that is the same as for the standard method, this can be easily increased to four by a suitable choice of the position of the stage abscissas. Secondly, the stability properties are massively enhanced. In particular, two versions of order four are A-stable, a fact which is quite unusual for explicit methods.     

Bifurcation and exact traveling wave solutions for dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution

In this article, we introduce the dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution in the sense of modified Riemann–Liouville derivative. We investigate the dynamical behavior, bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect. We apply the $(G^{\prime }∕G)$-expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave, kink-type wave, breaking wave and periodic wave solutions of the equation. Furthermore, the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures.     

Blow-up phenomena of solutions for a reaction–diffusion equation with weighted exponential nonlinearity

Blow-up phenomena for a reaction–diffusion equation with weighted exponential reaction term and null Dirichlet boundary condition are investigated. We establish sufficient conditions to guarantee existence of global solution or blow-up solution under appropriate measure sense by virtue of the method of super–sub solutions, the Bernoulli equation and the modified differential inequality techniques. Moreover, upper and lower bounds for the blow-up time are found in higher dimensional spaces and some examples for application are presented.     

Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg–de Vries models arising in dusty plasmas by using direct reduction method

In this paper, generalized models for both ($2+1$)-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and ($3+1$)-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both ($2+1$)-dimensional cmKdV and ($3+1$)-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach

In this paper, the solitary wave solutions of (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) equation are constructed which appear in the magnetized two-ion-temperature dusty plasma and quantum physics. Lie group of transformation method is proposed to investigate the solution of (3+1)-dimensional eZK equation via Lie symmetry method. The optimal system of one dimensional Lie subalgebra is constructed by using Lie point symmetries. The three dimensional eZK equation reduced into number of ordinary differential equations (ODEs) by applying similarity reductions. Consequently, solutions so extracted are more general than erstwhile known results. We have obtained twenty one solutions in the explicit form, some of them are likewise general and some are new for the best study of us. Eventually, single soliton, quasi-periodic soliton, multisoliton, lump-type soliton, traveling wave and solitary wave-interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results. Please check whether the affiliations are correct.     

Lump solutions to the Kadomtsev–Petviashvili I equation with a self-consistent source

Based on symbolic computations, lump solutions to the Kadomtsev–Petviashvili I (KPI) equation with a self-consistent source (KPIESCS) are constructed by using the Hirota bilinear method and an ansatz technique. In contrast with lower-order lump solutions of the Kadomtsev–Petviashvili (KP) equation, the presented lump solutions to the KPIESCS exhibit more diverse nonlinear phenomena. The method used here is more natural and simpler.     

The numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation

The main purpose of this paper is to give the numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation, considered in a recent paper (Cai, J. Complex. 25:420–436, 2009). This scheme leads to a fully discrete sparse linear system. We show that it requires a nearly linear computational cost to get this system, and the approximate solution of the resulting linear system preserves the optimal convergent order. Numerical experiments are presented to confirm the theoretical estimates.     

Compacton-like and kink-like waves for a higher-order wave equation of Korteweg–de Vries type

In this paper, the bifurcation method of the dynamical and numerical approach to differential equations to study higher order wave equations of Korteweg–de Vries (KdV) type is used. With this methodology we obtain the compacton-like and kink-like wave solutions of the high order KdV type equation. Their implicit expressions are given and their planar graphs are simulated. The results show that the numerical integrations are identical with the theoretical derivations.     

A new multi-symplectic scheme for the generalized Kadomtsev–Petviashvili equation

We propose a new scheme for the generalized Kadomtsev–Petviashvili (KP) equation. The multi-symplectic conservation property of the new scheme is proved. Back error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the KPI equation and the KPII equation are presented in detail.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.