Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method

Sepahvandzadeh, Amaneh; Ghazanfari, Bahman; Asadian, Nader

doi:10.3390/mca23040053

Cited by 2 publications

(2 citation statements)

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“…Khodabin et al [17] solved the stochastic initial value problems by using random Runge-Kutta methods of the second order, numerically. The numerical solution of stochastic generalized fractional diffusion equation via finite difference method together with its mean square convergence has been investigated in [18].…”

Section: Introductionmentioning

confidence: 99%

Simulating variable‐order fractional Brownian motion and solving nonlinear stochastic differential equations

Samadyar,

Ordokhani

2024

Math Methods in App Sciences

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Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for . Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm.

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Section: Introductionmentioning

confidence: 99%

Simulating variable‐order fractional Brownian motion and solving nonlinear stochastic differential equations

Samadyar,

Ordokhani

2024

Math Methods in App Sciences

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“…In thermal systems, the primary source of entropy generation is mass transfer, heat transfer, viscous dissipation, coupling among heat, electrical conduction, and chemical reaction, as examined in a pioneering series of publications by Bejan and co-workers [ 16 , 17 ]. Researchers have used various techniques for the solution of diffusion equations such as the Collocation method (CM) [ 18 ], Diffusion and Tsallis entropy [ 19 ], Entropy production, Symmetric fractional diffusion [ 20 ], Finite differences method in space-fractional diffusion equations [ 21 , 22 , 23 ], Homotopy analysis method (HAM) [ 24 ], Homotopy perturbation transform method (HPTM) [ 25 ] and Modified homotopy perturbation method (MHPM) [ 26 ], Mehshless method (MM) [ 27 ], One-Dimensional alpha fractional diffusion [ 28 ], Radial basis function method (RBFM) [ 29 ], and the Variational iteration method (VIM) [ 30 ].…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method

Shah

Khan

Mustafa

et al. 2019

Entropy

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In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.

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Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method

Cited by 2 publications

References 20 publications

Simulating variable‐order fractional Brownian motion and solving nonlinear stochastic differential equations

Simulating variable‐order fractional Brownian motion and solving nonlinear stochastic differential equations

Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method

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