The m-dimensional modified Helmholtz equation is considered and two relations between its solutions (called panharmonic functions) harmonic functions are established in a bounded domain. Both essentially rely on properties of the Newtonian potential. Other characteristics also obtained.

Abstract. In this study, we consider new finite difference schemes for solving the Helmholtz equation. Novel difference schemes which do not introduce truncation error are presented, consequently the exact solution for the Helmholtz equation can be computed numerically. The most important features of the new schemes are that while the resulting linear system has the same simple structure as tho...

Mean value properties of solutions to the m-dimensional Helmholtz and modified equations are considered. An elementary derivation these is given. It involves Euler–Poisson–Darboux equation. Despite similar form for both equations, their consequences distinguish essentially. The restricted mean property harmonic functions amended so that a function satisfying this in bounded domain special class...

In the applications of the method of fundamental solutions, locations of sources are treated either as variables or a priori known constants. In which, the former results in a nonlinear optimization problem and the other has to face the problem of locating sources. Theoretically, farther sources results in worse conditioning and better accuracy. In this paper, a practical procedure is provided ...

In this paper, the fractional partial differential equations are defined by modified Riemann-Liouville fractional derivative. With the help of fractional derivative and fractional complex transform, these equations can be converted into the nonlinear ordinary differential equations. By using solitay wave ansatz method, we find exact analytical solutions of the space-time fractional Zakharov Kuz...

The Quasi-Reversibility Regularization Method (Q-RRM) provides stable approximate solution of the Cauchy problem Helmholtz equation in Hilbert space by providing either additional information Laplace-type operator or imposed boundary conditions on equation. To help bridge this gap literature, a Modified Quasi-reversibility (MQ-RRM) is introduced to provide both occurring and equation, resulting...

with the same efficiency, but it turned out that this is a very difficult task. Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the He...