S. Karimi Vanani and A. Aminataei
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
Details
Keywords
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…
Abstract
Purpose
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.
Design/methodology/approach
Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method.
Findings
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].
Originality/value
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.
Details
Keywords
Mashallah Matinfar, Mostafa Eslami and Mohammad Saeidy
The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.
Abstract
Purpose
The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.
Design/methodology/approach
In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution.
Findings
In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems.
Originality/value
The basic idea described in this paper is expected to be further employed to solve other functional equations.
Details
Keywords
María José Pujol, Francisco A. Pujol, Fidel Aznar, Mar Pujol and Ramón Rizo
In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that…
Abstract
Purpose
In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.
Design/methodology/approach
In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.
Findings
Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.
Originality/value
The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.
Details
Keywords
Umesh and Manoj Kumar
The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size…
Abstract
Purpose
The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions.
Design/methodology/approach
First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed.
Findings
The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples.
Originality/value
Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.
Details
Keywords
The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the…
Abstract
Purpose
The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters.
Design/methodology/approach
To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models.
Findings
The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail.
Originality/value
From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.
Details
Keywords
Haiyan Zhang, Muhammad Nadeem, Asim Rauf and Zhao Guo Hui
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.
Abstract
Purpose
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.
Design/methodology/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.
Findings
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.
Originality/value
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.
Details
Keywords
Akbar Mohebbi, Mostafa Abbaszadeh and Mehdi Dehghan
The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two…
Abstract
Purpose
The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation.
Design/methodology/approach
In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives.
Findings
The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy.
Originality/value
The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.
Details
Keywords
The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave…
Abstract
Purpose
The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation.
Design/methodology/approach
The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica.
Findings
New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained.
Originality/value
The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.
Details
Keywords
Mohammad Heydari, Ghasem Barid Loghmani and Abdul-Majid Wazwaz
The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical…
Abstract
Purpose
The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.
Design/methodology/approach
This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.
Findings
Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.
Originality/value
In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.