In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald...In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald-Letnikov formula of order α ∈(0, 1). We investigate the stability analysis by using von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.展开更多
In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these things in mind in this manuscript we focus on the f...In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these things in mind in this manuscript we focus on the fractional Lagrangian and Harniltonian of the complex Bateman-Feshbach Tikochinsky oscillator. The numerical analysis of the corresponding fractionaJ Euler-Lagrange equations is given within the Griinwald-Letnikov approach, which is power series expansion of the generating function.展开更多
In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Griinwald-Letnikov...In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Griinwald-Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann-Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain trans-formations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model A comparative study between Griinwald-Letnikov method and Runge-Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.展开更多
In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the...In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pon- tryagin maximum principle. Two numerical methods are used to study the fractional- order optimal control problem. The methods are, the iterative optimal control method （OCM） and the generalized Euler method （GEM）. Positivity, boundedness and conver- gence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.展开更多
文摘In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald-Letnikov formula of order α ∈(0, 1). We investigate the stability analysis by using von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.
基金Supported in part by the Slovak Grant Agency for Science under Grants VEGA:1/0497/11,1/0746/11,1/0729/12the Slovak Research and Development Agency under Grant No.APVV-0482-11
文摘In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these things in mind in this manuscript we focus on the fractional Lagrangian and Harniltonian of the complex Bateman-Feshbach Tikochinsky oscillator. The numerical analysis of the corresponding fractionaJ Euler-Lagrange equations is given within the Griinwald-Letnikov approach, which is power series expansion of the generating function.
文摘In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Griinwald-Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann-Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain trans-formations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model A comparative study between Griinwald-Letnikov method and Runge-Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.
文摘In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pon- tryagin maximum principle. Two numerical methods are used to study the fractional- order optimal control problem. The methods are, the iterative optimal control method （OCM） and the generalized Euler method （GEM）. Positivity, boundedness and conver- gence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.
