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Thus, analytical evaluation of terms becomes very difficult or impossible. In order to keep away from that, in , the authors resort to use a new. Moreover, the Laplace transform and HPM is applied to solve the nonlinear FPEs in  which is required to apply the Laplace transform, construct a homotopy and solve the corresponding algebraic equations.
Furthermore,  Hesam et al. For numerical algorithms and solutions details can be found in .
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- The Fokker-Planck Equation Methods of Solution and Applications 18 2nd ed. 1996.
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The DJM has been extensively used by many researchers for the treatment of linear and nonlinear ordinary and partial differential equations of integer and fractional order, see [,26]. The method converges to the exact solution if it exists through successive approximations. However, for concrete problems, a few number of approximations can be used for numerical purposes with high degree of accuracy.
The DJM is simple to understand and easy to implement using computer packages and yields better results and does not require any restrictive assumptions for nonlinear terms as required by some existing techniques . It has been shown by many authors to be a powerful and reliable method to solve different kinds of nonlinear differential and integral equations.
AL-Jawary et al. In this paper, the applications of the DJM for both the linear and nonlinear Fokker-Planck equations in 1D, 2D and 3D will be presented to find the analytic solutions. Comparisons show that the DJM is more effective and convenient to use and overcomes the difficulties arising in existing techniques.
The present paper has been organized as follows. In Section 2 is devoted to the description of the Fokker-Planck equation. In Section 3 the basic idea of DJM and its convergence are illustrated. Finally in Section 6 the conclusion is presented. The Fokker and Planck for example, see [5,7,8,10] have first introduced the Fokker-Planck equation to represent the Brownian motion of particles. The equation of motion for the distribution function u x, t is given by:.
The coefficients B x and A x can also depend on time, i. In fact, it is a diffusion equation with an additional first order derivative with respect to x and it is also called forward Kolmogorov equation.
The Fokker-Planck Equation (2nd ed.)
On the other hand, the backward Kolmogorov equation is given in the following form :. The nonlinear FPE has important many applications in various sciences such as plasma physics, surface physics, population dynamic, biophysics, engineering, neuro sciences, nonlinear hydrodynamics, polymer physics, laser physics, pattern formation, psychology and marketing , which is for one variable in the form:. It can also be clearly seen that Eq.
B and f is a known function [,26]. We are looking for a solution u of Eq. The condition for convergence of the series ui is presented below. One can find more details in . Theorem 3. Let us assume the inverse operator Lt-1 exists and it can be take with respect t from 0 to t, i. By applying the DJM for Eq. To verify the convergence of the methods, we applied the method to some test problems for which an analytical solutions are available. Example 3. Consider the 1D linear backward Kol-mogorov equation in Eq. According to the DJM, the following terms can be achieved:.
The Fokker-Planck Equation: Methods of Solution and Applications
According to the DJM, we obtain the following components:. By using the DJM, the following recursive relation is obtained:.
In this paper, the efficient iterative method namely NIM or DJM is implemented to obtain an analytical solutions for solving 1D, 2D, 3D linear and nonlinear Fokker-Planck equations using the initial condition only. The DJM is simple to understand and easy to implement and does not require any restrictive assumptions for nonlinear terms as required by some existing techniques.
Moreover, by solving some examples, it is seems that the DJM appears to be very accurate to employ with reliable results. The results obtained in this paper confirm that the DJM is a powerful and efficient technique for obtaining the exact solutions for 1D, 2D, 3D linear and nonlinear Fokker-Planck equations which have great significance in many different fields of science and engineering.
Furthermore, there is less computation needed in comparison with the Adomian decomposition method, the homotopy perturbation method and variational iteration method. The author would like to thank the anonymous referees and Editor in Chief for their valuable suggestions. Two analytical method for time -fractional nonlinear coupled Boussinesq-Burger equations arises in propagation of shallow water waves.
Nonlinear Dyn ; Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv Nonlinear Anal Zoom Zoom. Availability Usually despatched within 2 weeks. With Free Saver Delivery. Facebook Twitter Pinterest Share. Description Reviews More Details.
Description This is the first textbook to include the matrix continued-fraction method, which is very effective in dealing with simple Fokker-Planck equations having two variables. Other methods covered are the simulation method, the eigen-function expansion, numerical integration, and the variational method. Not in stock at the moment, but more on the way. Add to basket. Add to Wish List. Share on Facebook. This is the first textbook to include the matrix continued-fraction method, which is very effective in dealing with simple Fokker-Planck equations having two variables.
Other methods covered are the simulation method, the eigen-function expansion, numerical integration, and the variational method. Each solution is applied to the statistics of a simple laser model and to Brownian motion in potentials. The whole is rounded off with a supplement containing a short review of new material together with some recent references.
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