The extended tanh method for solving systems of nonlinear wave equations

The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

Stepanov ergodic perturbations for some neutral partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of μ?pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ?pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 John Wiley & Sons, Ltd.     

EPSILON-ALGORITHM AND ETA-ALGORITHM OF GENERALIZED INVERSE FUNCTION-VALUED PADE APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

Two efficient recursive algorithms epsilon- algorithm and eta-algorithm are approximants were used to accelerate the convergence of the power series with functionvalued coefficients and to estimate characteristic value of the integral equations. Famous two algorithms.     

NUMERICAL SIMULATION OF SHALLOW WATER WAVE PROPAGATION USING A BOUNDARY ELEMENT WAVE EQUATION MODEL

The present paper makes use of a wave equation formulation of the primitive shallow water equations to simulate one-dimensional free surface flow. A numerical formulation of the boundary element method is then developed to solve the wave continuity equation using a time-dependent fundamental solution, while an explicit finite difference scheme is used to derive velocities from the primitive momentum equation. One-dimensional free surface flows in open channels are treated and the results compared with analytical and numerical solutions. © 1997 John Wiley & Sons, Ltd.     

On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree

Formulas for calculating Legendre functions and associated Legendre functions of the first kind of complex degree using an Ermakov—Lewis invariant are presented. These formulas are straight-forward to implement numerically and are motivated by the lack of computational routines in standard university tools like those of MatLab and Maple. Angular waves propagating in opposite directions are also obtained. The results are particularly useful in complex angular momentum theories and nearside/farside analysis of spin-dependent angular scattering from central potentials.     

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

This paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.