An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

A Robin boundary value problem for the Cauchy–Riemann operator in a ring domain

In this paper, we study a Robin condition for the inhomogeneous Cauchy–Riemann equation w z ¯ = f {w_{\bar{z}}=f} in a ring domain R, by reformulating it as a Dirichlet boundary condition.

On the extremal solutions of superlinear Helmholtz problems

In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

Boundary value problem for the four-dimensional Gellerstedt equation

In this work, the solvability of the problem with Neumann and Dirichlet boundary conditions for the Gellerstedt equation in four variables is investigated. The energy integral method is used to prove the uniqueness of the solution to the problem. In addition to it, formulas for differentiation, autotransformation, and decomposition of hypergeometric functions are applied. The solution is obtained explicitly and is expressed by Lauricella’s hypergeometric function.

Accuracy Investigation of FDM, FEM and MoM for a Numerical Solution of the 2D Laplace’s Differential Equation for Electrostatic Problems

Laplace’s differential equation is one of the most important equations which describe the continuity of a system in various fields of engineering. As a system gets more complex, the need for solving this equation numerically rises. In this paper we present an accuracy investigation of three of the most significant numerical methods used in computational electromagnetics by applying them to solve Laplace’s differential equation in a two-dimensional domain with Dirichlet boundary conditions. We investigate the influence of discretization on the relative error obtained by applying each method. We point out advantages and disadvantages of the investigated computational methods with emphasis on the hardware requirements for achieving certain accuracy.