scholarly journals The Function Inverfc ?

1960 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
JR Phllip
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Independent Variable ◽  
General Method ◽  
Diffusion Problems

The function inverfc 6 arises in certain diffusion problems when concentration is taken as an independent variable. It enters into a general method of exact solution of the concentration-dependent diffusion equation. An account is given of the properties of this function, and of its derivatives and integrals. The function

scholarly journals General Method of Exact Solution of the Concentration?Dependent Diffusion Equation

1960 ◽  
Vol 13 (1) ◽  
pp. 1 ◽  
Author(s):  
JR Philip
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Exact Solutions ◽  
General Method

Only three forms of D(6) have previously been known to yield exact solutions of the equation

scholarly journals Rational Krylov methods for fractional diffusion problems on graphs

2021 ◽  
Author(s):  
Michele Benzi ◽  
Igor Simunec
Keyword(s):  
Analytic Function ◽  
Diffusion Equation ◽  
Graph Laplacian ◽  
Fractional Diffusion ◽  
Singular Matrix ◽  
Krylov Methods ◽  
Fractional Powers ◽  
Directed Networks ◽  
Rank One ◽  
Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

scholarly journals Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Shrödinger Equation with Complicated Potential

2021 ◽  
Author(s):  
Michael I. Tribelsky
Keyword(s):  
Ground State ◽  
Nonlinear Diffusion ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
Dinger Equation ◽  
Bottom Part ◽  
Diffusion Type ◽  
Traveling Pulses ◽  
General Method ◽  
Diffusion Problems

The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically the problem is reduced to the calculation of the "energy" of the ground state in Schrödinger equation with a complicated potential. A general method to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells is proposed and applied. Possible generalization of the approach to other types of nonlinear diffusion equations is discussed.

COMPARISON BETWEEN GLOBAL, CLASSICAL DOMAIN DECOMPOSITION AND LOCAL, SINGLE AND DOUBLE COLLOCATION METHODS BASED ON RBF INTERPOLATION FOR SOLVING CONVECTION-DIFFUSION EQUATION

2008 ◽  
Vol 19 (11) ◽  
pp. 1737-1751 ◽  
Author(s):  
GAIL GUTIERREZ ◽  
WHADY FLOREZ
Keyword(s):  
Diffusion Equation ◽  
Domain Decomposition ◽  
Mixed Boundary ◽  
Domain Decomposition Method ◽  
Collocation Methods ◽  
Special Treatment ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Full Domain ◽  
Diffusion Problems

This work presents a performance comparison of several meshless RBF formulations for convection-diffusion equation with moderate-to-high Peclet number regimes. For the solution of convection-diffusion problems, several comparisons between global (full-domain) meshless RBF methods and mesh-based methods have been presented in the literature. However, in depth studies between new local RBF collocation methods and full-domain symmetric RBF collocation methods are not reported yet. The RBF formulations included: global symmetric method, symmetric double boundary collocation method, additive Schwarz domain decomposition method (DDM) when it is incorporated into two anterior approaches, and local single and double collocation methods. It can be found that the accuracy of solutions deteriorates as Pe increases, if no special treatment is used. From the numerical tests, it seems that the local methods, especially the derived double collocation technique incorporating PDE operator, are more effective than full domain approaches even with iterative DDM in solving moderate-to-high Pe convection-diffusion problems subject to mixed boundary conditions.

On the Neutron multi-group kinetic diffusion equation in a heterogeneous slab: An exact solution on a finite set of discrete points

2015 ◽  
Vol 76 ◽  
pp. 271-282 ◽  
Author(s):  
Celina Ceolin ◽  
Marcelo Schramm ◽  
Marco T. Vilhena ◽  
Bardo E.J. Bodmann
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Finite Set ◽  
Kinetic Diffusion

Stepwise regularization method for a nonlinear Riesz–Feller space-fractional backward diffusion problem

2019 ◽  
Vol 27 (6) ◽  
pp. 759-775
Author(s):  
Dang Duc Trong ◽  
Dinh Nguyen Duy Hai ◽  
Nguyen Dang Minh
Keyword(s):  
Exact Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Convergence Estimate ◽  
Nonlinear Source ◽  
Space Fractional Diffusion Equation ◽  
Final Data

Abstract In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

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