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A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions
Jyh-Haw Tang,
Chao-Kang Feng
Issue:
Volume 11, Issue 2, April 2022
Pages:
38-47
Received:
10 March 2022
Accepted:
2 April 2022
Published:
9 April 2022
Abstract: Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.
Abstract: Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a...
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Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems
Khaled Abdalla Ishag,
Mohamed Ahmed Abdallah,
Abulfida Mohamed Ahmed
Issue:
Volume 11, Issue 2, April 2022
Pages:
48-55
Received:
22 March 2022
Accepted:
11 April 2022
Published:
20 April 2022
Abstract: This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α1=α2=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α1=0.5 and α2=0.5 together with non-fractional derivative index α1=1 and α2=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.
Abstract: This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the su...
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A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis
Issue:
Volume 11, Issue 2, April 2022
Pages:
56-59
Received:
20 March 2022
Accepted:
6 April 2022
Published:
25 April 2022
Abstract: Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].
Abstract: Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed...
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