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Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.

This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.

This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.

This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional derivatives. Fokker‐Planck equation with positive integer order is also considered.

The proposed algorithm converts the desired FFPE to a set of algebraic equations using orthogonal polynomials as basis functions. The paper states some concepts, properties and advantages of proposed algorithm and its applications for solving FFPE.

Some illustrative numerical experiments including linear and nonlinear FFPE are given and some comparisons are made between OTM and variational iteration method, Adomian decomposition method and homotpy perturbation method.

Results demonstrate some capabilities of the proposed algorithm such as the simplicity, the accuracy and the convergency. Also, this is the first presentation of this algorithm for FFPE.

The purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.

First, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.

It is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.

Although many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.

This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.

By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).

This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.

The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary conditions.

Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method.

The accuracy of the proposed scheme is demonstrated on three test examples. The numerical solutions prove that the proposed method is reliable and yields compatible results with the exact solutions. The scheme provides better results than the schemes [9, 14].

The developed scheme is a new scheme for Fokker-Planck equations. The scheme based on Haar wavelets is expended for nonlinear partial differential equations with variable coefficients.

The current analysis produces the fractional sample of non-Newtonian Casson and Williamson boundary layer flow considering the heat flux and the slip velocity. An extended sheet with a nonuniform thickness causes the steady boundary layer flow’s temperature and velocity fields. Our purpose in this research is to use Akbari Ganji method (AGM) to solve equations and compare the accuracy of this method with the spectral collocation method.

The trial polynomials that will be utilized to carry out the AGM are then used to solve the nonlinear governing system of the PDEs, which has been transformed into a nonlinear collection of linked ODEs.

The profile of temperature and dimensionless velocity for different parameters were displayed graphically. Also, the effect of two different parameters simultaneously on the temperature is displayed in three dimensions. The results demonstrate that the skin-friction coefficient rises with growing magnetic numbers, whereas the Casson and the local Williamson parameters show reverse manners.

Moreover, the usefulness and precision of the presented approach are pleasing, as can be seen by comparing the results with previous research. Also, the calculated solutions utilizing the provided procedure were physically sufficient and precise.

The purpose of this study is to develop a comprehensive size-dependent piezoelectric thermo-viscoelastic coupling model that accounts for two fundamentally distinct size-dependent models that govern fractional dual-phase lag heat transfer and viscoelastic deformation, respectively.

The fractional calculus has recently been shown to capture precisely the experimental effects of viscoelastic materials. The governing equations are combined into a unified system, from which certain theorems results on linear coupled and generalized theories of thermo-viscoelasticity may be easily established. Laplace transforms and state–space approach will be used to determine the generic solution when any set of boundary conditions exists. The derived formulation is used to two concrete different problems for a piezoelectric rod. The numerical technique for inverting the transfer functions is used to generate observable numerical results.

Some analogies of impacts of nonlocal thermal conduction, nonlocal elasticity and DPL parameters as well as fractional order on thermal spreads and thermo-viscoelastic response are illustrated in the figures.

The results in all figures indicate that the nonlocal thermal and viscoelastic parameters have a considerable influence on all field values. This discovery might help with the design and analysis of thermal-mechanical aspects of nanoscale devices.

To define the main elements of a formal calculus which deals with fractional Brownian motion (fBm), and to examine its prospects of applications in systems science.

The approach is based on a generalization of the Maruyama's notation. The key is the new Taylor's series of fractional order f(x+h)=E_α(h^αD^α)f(x), where E_α( · ) is the Mittag‐Leffler function.

As illustrative applications of this formal calculus in systems science, one considers the linear quadratic Gaussian problem with fractal noises, the analysis of the equilibrium position of a system disturbed by a local fractal time, and a model of growing which involves fractal noises. And then, one examines what happens when one applies the maximum entropy principle to systems involving fBms (or shortly fractals).

The framework of this paper is applied mathematics and engineering mathematics, and the results so obtained allow the practical analysis of stochastic dynamics subject to fractional noises.

The direct prospect of application of this approach is the analysis of some stock markets dynamics and some biological systems.

The fractional Taylor's series is new and thus so are all its implications.

The purpose of this paper is to develop an alternative numerical approach for describing fractional diffusion in Cartesian and non‐Cartesian domains using a Monte Carlo random walk scheme. The resulting domain shifting scheme provides a numerical solution for multi‐dimensional steady state, source free diffusion problems with fluxes expressed in terms of Caputo fractional derivatives. This class of problems takes account of non‐locality in transport, expressed through parameters representing both the extent and direction of the non‐locality.

The method described here follows a similar approach to random walk methods previously developed for normal (local) diffusion. The key differences from standard methods are: first, the random shifting of the domain about the point of interest with, second, shift steps selected from non‐symmetric, power‐law tailed, Lévy probability distribution functions.

The domain shifting scheme is verified by comparing predictive solutions to known one‐dimensional and two‐dimensional analytical solutions for fractional diffusion problems. The scheme is also applied to a problem of fractional diffusion in a non‐Cartesian annulus domain. In contrast to the axisymmetric, steady state solution for normal diffusion, a non‐axisymmetric solution results.

This is the first random walk scheme to utilize the concept of allowing the domain to undergo the random walk about a point of interest. Domain shifting scheme solutions of fractional diffusion in non‐Cartesian domains provide an invaluable tool to direct the development of more sophisticated grid based finite element inspired fractional diffusion schemes.