Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

A three-dimensional（3D） parabolic equation（PE） model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

CHEBYSHEV PSEUDOSPECTRAL-HYBRID FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION

In this paper,Chebyshev pseudospectral-finite element schemes are proposed for solving three dimensional vorticity equation.Some approximation results in nonisotropic Sobolev spaces are given.The generalized stability and the convergence are proved strictly.The numerical results show the advantages of this method.The technique in this paper is also applicable to other three-dimensional nonlinear problems in fluid dynamics.

FOURIER-CHEBYSHEV PSEUDOSPECTRAL METHOD FOR THREE-DIMENSIONAL VORTICITY EQUATION WITH UNILATERALLY PERIODIC BOUNDARY CONDITION

A Fourier-Chebyshev pseudospectral scheme is proposed for three-dimensionalvorticily equation with unilaterally periodic boundary condition. The generalized stability and convergence are analysed. The numerical results are presented.

INTERFACIAL CRACK ANALYSIS IN THREE-DIMENSIONAL TRANSVERSELY ISOTROPIC BI-MATERIALS BY BOUNDARY INTEGRAL EQUATION METHOD

The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropic materials.

OpenMP-Based PCG Solver for Three-Dimensional Heat Equation

As one of the most important mathematics-physics equations, heat equation has been widely used in engineering area and computing science research. Large-scale heat problems are difficult to solve due to computational intractability. The parallelization of heat equation is available to improve the simulation model efficiency. In order to solve the three-dimensional heat problems more rapidly, the OpenMP was adopted to parallelize the preconditioned conjugate gradient （PCG） algorithm in this paper. A numerical experiment on the three-dimensional heat equation model was carried out on a computer with four cores. Based on the test results, it is found that the execution time of the original serial PCG program is about 1.71 to 2.81 times of the parallel PCG program executed with different number of threads. The experiment results also demonstrate the available performance of the parallel PCG algorithm based on OpenMP in terms of solution quality and computational performance.

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

A NEW HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THREE-DIMENSIONAL PARABOLIC EQUATIONS

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).

Symmetry Reductions and Explicit Solutions for a Generalized Zakharov-Kuznetsov Equation

Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equationare explored by the method of the improved generalized auxiliary differential equation.Many explicit analytic solutionsof the Z-K equation are obtained.The methods used to solve the Z-K equation can be employed in further work toestablish new solutions for other nonlinear partial differential equations.

Explicit multi-symplectic method for the Zakharov-Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.

New Explicit Solutions of the Generalized (2 + 1)-Dimensional Zakharov-Kuznetsov Equation

his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Symmetry reduction and exact solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation

By means of the classical method, we investigate the （3＋1）-dimensional Zakharov-Kuznetsov equation. The symmetry group of the （3＋1）-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the （3＋1）-dimensional Zakharov-Kuznetsov equation.

Similarity Solutions for Generalized Variable Coefficients Zakharov-Kuznetsov Equation under Some Integrability Conditions

In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov- Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal system seven basic fields are determined, and for every vector field in the optimal system the admissible forms of the coefficients are found and this also leads us to transform the given equation into partial differential equations in two variables. After using some referenced transformations the mentioned partial differential equations eventually reduce to ordinary differential equations. The search for solutions to those equations has yielded many exact solutions in most cases.

Solitary Wave Solutions and Rational Solutions for Modified Zakharov-Kuznetsov Equation with Initial Value Problem

The modified Zakharov-Kuznetsov equation with the initial value problem is studied numerically by means of homotopy perturbation method. The analytical approximate solutions of the modified Zakharov-Kuznetsov equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.

Analytical Approach to Fractional Zakharov-Kuznetsov Equations by He's Homotopy Perturbation Method

The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.

Construction of doubly-periodic solutions to nonlinear partial differential equations using improved Jacobi elliptic function expansion method and symbolic computation

Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.