scholarly journals Vector Additive Decomposition for 2D Fractional Diffusion Equation

2008 ◽  
Vol 13 (2) ◽  
pp. 137-143
Author(s):  
N. Abrashina-Zhadaeva ◽  
N. Romanova
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Analytical Solution ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Finite Difference Approximation ◽  
Partial Differential

Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations.  An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients.  In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.

Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

2016 ◽  
Vol 19 (3) ◽  
pp. 632-647 ◽  
Author(s):  
John C. Morrison ◽  
Kyle Steffen ◽  
Blake Pantoja ◽  
Asha Nagaiya ◽  
Jacek Kobus ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Domain Decomposition ◽  
Diatomic Molecule ◽  
Linear Equations ◽  
Finite Difference Approximation ◽  
Partial Differential ◽  
Hartree Fock ◽  
Finite Difference Approach

AbstractIn order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.

scholarly journals Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 335 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Muhammad Arif ◽  
Poom Kumam
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Third Order ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
In Series

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

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