scholarly journals Numerical Treatments for Nonlinear Integro-Fractional Differential Equations of Volterra-Hammerstein Type using Runge-Kutta Method with the aid of Finite Difference Approximation

2021 ◽  
Vol 23 (2) ◽  
pp. 129-146
Author(s):  
Shazad Shawki Ahmed ◽  
◽  
Mariwan Rashid Ahmed ◽  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Fractional Differential ◽  
Runge Kutta ◽  
Numerical Treatments

NEW METHOD FOR CHARACTERISTIC EVOLUTIONS IN NUMERICAL RELATIVITY

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
ZHOUJIAN CAO
Keyword(s):  
Cauchy Problem ◽  
Finite Difference ◽  
Differential Equations ◽  
Numerical Relativity ◽  
Einstein Equations ◽  
New Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Special Difference

The main task of numerical relativity is to solve Einstein equations with the aid of supercomputer. There are two main schemes to write Einstein equations explicitly as differential equations. One is based on 3 + 1 decomposition and reduces the Einstein equations to a Cauchy problem. The another takes the advantage of the characteristic property of Einstein equations and reduces them to a set of ordinary differential equations. The latter scheme is called characteristic formalism which is free of constraint equations in contrast to the corresponding Cauchy problem. Till now there is only one well developed code (PITT code) for characteristic formalism. In PITT code, special finite difference algorithm is adopted for the numerical calculation. And it is this special difference algorithm that restricts the numerical accuracy order to second-order. In addition, this special difference algorithm makes the popular Runge–Kutta method used in Cauchy problem not available. In this paper, we modify the equations for characteristic formalism. Based on our new set of equations, we can use usual finite difference method as done in usual Cauchy evolution. And Runge–Kutta method can also be adopted naturally. We develop a set of code in the framework of AMSS-NCKU code based on our new method and some numerical tests are done.

scholarly journals Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

2020 ◽  
Vol 14 (3) ◽  
pp. 378-388
Author(s):  
Esther Y. Bunga ◽  
Meksianis Z. Ndii
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Kutta Method ◽  
System Of Differential Equations ◽  
Time Step ◽  
Differential Transformation ◽  
Runge Kutta Method ◽  
Runge Kutta

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM)  to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

scholarly journals A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Xianjuan Li ◽  
Yanhui Su
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Spectral Collocation ◽  
Parallel Method ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

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