Dimensional Nonlinear Partial Differential Equations Research Articles

New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications

In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by utilizing modified extended direct algebraic method. Soliton solutions in different forms such as bell and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc are obtained, which have large applications in different branches of physics and other areas of applied sciences. The obtained solutions are also presented graphically. Furthermore, many other nonlinear evolution equations arising in mathematical physics and engineering can also be solved by this powerful, reliable and capable method. The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica equation and (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the reliability and effectiveness of the current method.

Lump solutions to dimensionally reduced Kadomtsev–Petviashvili-like equations

In this paper, we present a new (3+1)-dimensional KP-like nonlinear partial differential equation and construct the lump solutions, rationally localized in all directions in the space, to its two dimensionally reduced cases. The proposed method in this work is based on a generalized bilinear differential equation, which implies that we can build the lump solutions to the presented KP-like equation from special polynomial solutions to the aforementioned generalized bilinear equation. Since there are totally six free parameters in the resulting lump solutions, we can get the sufficient and necessary conditions guaranteeing analyticity and rational localization of the solutions by using these six free parameters. Two special cases are plotted as illustrative examples, and some contour plots with different determinant values are presented to show that the corresponding lump solution tends to zero when the determinant approaches zero.

Rational solutions to two new KP-like equations

In this paper, we will present a new (3+1)-dimensional KP-like nonlinear partial differential equation, and construct the rational solutions to this new (3+1)-dimensional KP-like equation. The proposed method in this work is based on a generalized bilinear differential equation, which implies that we can build the rational solutions to the presented KP-like equation from polynomial solutions to the aforementioned generalized bilinear equation. Moreover, for y=z, this new model reduces to the new (2+1)-dimensional KP-like nonlinear partial differential equation, which will be studied as well herein.

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.

Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

Abstract In this paper, the generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (VCKPE), which can describe nonlinear phenomena in fluids or plasmas, is studied by using two different Clarkson and Kruskal (CK) direct methods, namely, the classical CK and the modified enlarged CK method. A similarity reduction to a (2+1)-dimensional nonlinear partial differential equation and a direct similarity reduction to a nonlinear ordinary differential equation are obtained, respectively. By solving the reduced ordinary differential equation, new solitary, periodic, and singular solutions for the VCKPE are obtained. Some figures for the soliton and periodic wave solutions are given to reflect the effect of the variable coefficients on the solution propagation. Finally, the comparison between the two different CK techniques indicates that the modified enlarged CK technique is clearly more powerful and simple than the classical CK technique.

The compound [formula omitted]-expansion method and double non-traveling wave solutions of (2+1) -dimensional nonlinear partial differential equations

To seek the exact double non-traveling wave solutions of nonlinear partial differential equations, the compound (G′G)-expansion method is firstly proposed in this paper. With the aid of symbolic computation, this new method is applied to construct double non-traveling wave solutions of (2+1)-dimensional Painlevé integrable Burgers equation and (2+1)-dimensional breaking soliton equation. As a result, abundant double non-traveling wave solutions including double hyperbolic function solutions, double trigonometric function solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via the proposed method. These exact solutions contain arbitrary functions, which may be helpful to explain some complex phenomena. When the parameters are taken as special values, the double solitary-like wave solutions can be derived from double hyperbolic function solutions. Furthermore, the time evolutions of double solitary-like wave solutions are discussed in detail.

Feng’s First Integral Method for Analytic Treatment of Two Higher Dimensional Nonlinear Partial Differential Equations

Analytic treatment of nonlinear partial differential equations is increasingly attracting the attention of academic researchers worldwide. Therefore, in the present study, Feng’s first integral method which is based on the ring theory of commutative algebra is utilized for analytic treatment of two higher dimensional nonlinear partial differential equations, including the (3+1)-dimensional modified KdV–Zakharov–Kuznetsev equation and the (3+1)-dimensional Zakharov–Kuznetsov equation.

Solutions of multidimensional partial differential equations representable as a one-dimensional flow

We propose an algorithm for reducing an (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow ut + \(w_{x_1 } \) (u, ux uxx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.

Particular solutions to multidimensional PDEs with KdV-type nonlinearity

We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation (ODE). In a simplest case, $n=1$, the ODE is solvable in terms of elementary functions. Next choice, $n=2$, yields the cnoidal waves for the special case of Zakharov-Kuznetsov equation. The proposed method is based on the deformation of the characteristic of the equation $u_t-uu_{x_1}=0$ and might also be useful in study the higher dimensional PDEs with arbitrary linear part and KdV-type nonlinearity (i.e. the nonlinear term is $u_{x_1} u$).

Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation

Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of(2+1)-dimensional nonlinear partial differential equation. The resulting theory is applied to the(2+1)-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.

Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients

The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions oft. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.

Bell Waves and Kind Waves for a (1+1)-Dimensional Nonlinear Partial Differential Equation

With the help of the symbolic computation system Maple and the mapping approach and a linear variable separation approach, a new family of exact solutions of the (1+1)-dimensional Burgers system is derived. Based on the derived solitary wave solution, some novel bell wave and kind wave excitations are investigated.

A Novel Numerical Method of for Three-Dimensional Non-Linear Triharmonic Equations

AbstractIn this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values ofu,∂2u/∂n2 and∂4u/∂n4are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods.

HIGH ACCURACY ARITHMETIC AVERAGE TYPE DISCRETIZATION FOR THE SOLUTION OF TWO-SPACE DIMENSIONAL NONLINEAR WAVE EQUATIONS

In this paper, we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyperbolic partial differential equation of the form utt = A(x, y, t)uxx + B(x, y, t)uyy + g(x, y, t, u, ux, uy, ut), 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions. We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation. The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable. Numerical results are provided to illustrate the usefulness of the proposed method.

New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp‐Function Method

We construct new analytical solutions of the (3 + 1)‐dimensional modified KdV‐Zakharov‐Kuznetsev equation by the Exp‐function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp‐function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher‐dimensional nonlinear partial differential equation.

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.

A modification of the method of characteristics: A new class of multidimensional partially integrable nonlinear systems

We represent an algorithm reducing a big class of systems of (M+1)-dimensional nonlinear partial differential equations (PDEs) to the systems of M-dimensional first order PDEs. Thus, we integrate the original system with respect to only one independent variable reducing its dimensionality by one. For this reason we call such systems partially integrable ones. In particular, if M=1, then M-dimensional PDEs become ODEs. Our method may be referred to as a generalization of the method of characteristics. Possible application in hydrodynamics is discussed.

Hyperelliptic functions solutions of some nonlinear partial differential equations using the direct method

In this paper, we will use a simple and direct method to obtain particular solutions of some (2+1) dimensional nonlinear partial differential equations expressed in terms of the Kleinian hyperelliptic functions for a given curve y 2 = f ( x ) whose genus is two. We observe that this method generalizes the auxiliary method.

A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach

AbstractIn this article, using coupled approach, we discuss fourth order finite difference approximation for the solution of two dimensional nonlinear biharmonic partial differential equations on a 9‐point compact stencil. The solutions of unknown variable and its Laplacian are obtained at each internal grid points. This discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions. We require only system of two equations to obtain the solution and its Laplacian. The proposed fourth order method is used to solve a set of test problems and produce high accuracy numerical solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

On Periodic Wave Solutions to (1+1)-Dimensional Nonlinear Physical Models Using the Sine-Cosine Method

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.