One class of MHD equations: Conservation laws and exact solutions

The paper analyzes one of the models of equations of magnetohydrodynamics (MHD) derived earlier. The model was obtained as a result of group classification of the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. The use of Lagrangian coordinates made it possible to solve four equations, which led to the form of reduced equations containing four arbitrary functions: entropy and a three-dimensional vector associated with the magnetic field. The objective of this work is to develop conservation laws and exact solutions for the model. Conservation laws are obtained using Noether's theorem, while exact solutions are obtained either explicitly or by solving a system of ordinary or partial differential equations with two independent variables. Numerical methods are employed for the latter solutions.     

Dynamics and exact solutions of non-evolutionary partial differential equations

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential equations by B. Kruglikov, O. Lychagina and V. Lychagin. This theory is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of PDEs. The proposed method is used to construct exact particular solutions of linear differential equations (Helmholtz equations and equations of variable type).     

Conservation laws and exact solutions of the Whitham-type equations

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.     

Group classifications, optimal systems and exact solutions to the generalized Thomas equations

In this paper, the complete group classifications are performed on the types of Thomas equations (TEs), which arise in the study of chemical exchange progress, etc., all of the vector fields of the equations are presented. Then, the optimal system of the general Thomas equation is given, and all of the symmetry reductions and exact solutions generated from the optimal system are investigated. Furthermore, the exact analytic solutions to the Thomas equations are obtained by the generalized power series method.     

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.     

Some exact solutions to Stefan problems with fractional differential equations

Some exact solutions to the first, second and extended Stefan problems with fractional time derivative described in the Caputo sense are given by means of fractional Green's function and Wright function in this paper. By the aid of simple calculations, many results of differential equations of integer order can be obtained as special cases of the results given by this paper.     

A class of large solutions to the 3D incompressible MHD and Euler equations with damping

In this paper, we derive the global existence of smooth solutions of the 3 D incompressible Euler equations with damping for a class of laxge initial data, whose Sobolev norms H~s can be arbitrarily large for any s ≥ 0. The approach is through studying the quantity representing the difference between the vorticity and velocity. And also, we construct a family of large solutions for MHD equations with damping.     

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

New exact solutions of differential equations derived by fractional calculus

Fractional calculus generalizes the derivative and antiderivative operations dⁿ/dzⁿ of differential and integral calculus from integer orders n to the entire complex plane. Methods are presented for using this generalized calculus with Laplace transforms of complex-order derivatives to solve analytically many differential equations in physics, facilitate numerical computations, and generate new infinite-series representations of functions. As examples, new exact analytic solutions of differential equations, including new generalized Bessel equations with complex-power-law variable coefficients, are derived.     

Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations

In this paper, the Lie symmetry analysis is performed on the fifth-order KdV types of equations which arise in modeling many physical phenomena. The similarity reductions and exact solutions are obtained based on the optimal system method. Then the exact analytic solutions are considered by using the power series method.     

Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

Symmetry reduction, exact solutions and conservation laws of the Sawada-Kotera-Kadomtsev-Petviashvili equation

In this paper, by applying a direct symmetry method, we obtain the symmetry reduction, group invariant solution and many new exact solutions of SK-KP equation, which include Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and so on. At last, we also give the conservation laws of SK-KP equation.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in $H^s$.     

A new algebraic procedure to construct exact solutions of nonlinear differential-difference equations

This paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential-difference equations. The discrete sine-Gordon equation and differential-difference asymmetric Nizhnik-Novikov-Veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure, where various types of exact travelling wave solutions for these nonlinear differential-difference equations have been constructed. It is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential-difference equations.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

A reduction method for semilinear elliptic equations and solutions concentrating on spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A⊆R^2m$A\subseteq {\mathbb{R}}^{2m}$, m?2$m?2$, invariant by the action of a certain symmetry group can be reduced to a nonhomogeneous similar problem in an annulus D⊂R^m+1$D\subset {\mathbb{R}}^{m+1}$, invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A   which concentrate on one or two (m−1)$(m-1)$ dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.     

New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.