Abstract: In this paper, the techniques to solve time dependent electromagnetic wave propagation equations based on the Finite Difference Method (FDM) are proposed by comparing the results with Finite Element Method (FEM) in 2D while discussing some special simulation examples. Here, 2D dynamical wave equations for lossy media, even with a constant source, are discussed for establishing symbolic manipulation of wave propagation problems. The main objective of this contribution is to introduce a comparative study of two suitable numerical methods and to show that both methods can be applied effectively and efficiently to all types of wave propagation problems, both linear and nonlinear cases, by using symbolic computation. However, the results show that the FDM is more appropriate for solving the nonlinear cases in the symbolic solution. Furthermore, some specific complex domain examples of the comparison of electromagnetic waves equations are considered. Calculations are performed through Mathematica software by making some useful contribution to the programme and leveraging symbolic evaluations of FEM and FDM.
Abstract: In this paper a class of numerical methods to solve linear and nonlinear PDEs and also systems of PDEs is developed. The Differential Transform method associated with the Method of Lines (MoL) is used. The theory for linear problems is extended to the nonlinear case, and a recurrence relation is established. This method can achieve an arbitrary high-order accuracy in time. A variable stepsize algorithm and some numerical results are also presented.
Abstract: We provide a maximum norm analysis of a finite
element Schwarz alternating method for a nonlinear elliptic boundary
value problem of the form -Δu = f(u), on two overlapping sub
domains with non matching grids. We consider a domain which is
the union of two overlapping sub domains where each sub domain
has its own independently generated grid. The two meshes being
mutually independent on the overlap region, a triangle belonging to
one triangulation does not necessarily belong to the other one. Under
a Lipschitz assumption on the nonlinearity, we establish, on each sub
domain, an optimal L∞ error estimate between the discrete Schwarz
sequence and the exact solution of the boundary value problem.
Abstract: The objective of this paper is to use the Pfaffian
technique to construct different classes of exact Pfaffian solutions and
N-soliton solutions to some of the generalized integrable nonlinear
partial differential equations in (3+1) dimensions. In this paper, I will
show that the Pfaffian solutions to the nonlinear PDEs are nothing but
Pfaffian identities. Solitons are among the most beneficial solutions
for science and technology, from ocean waves to transmission of
information through optical fibers or energy transport along protein
molecules. The existence of multi-solitons, especially three-soliton
solutions, is essential for information technology: it makes possible
undisturbed simultaneous propagation of many pulses in both directions.