# e-book Fractional Analysis: Methods of Motion Decomposition

Moreover, phenomena in electromagnetics, acoustics, viscoplasticity, electrochemistry are also described by differential equations of fractional order Podlubny, ; Shawagfeh, The Fokker-Planck F-P equation is one of the basic equations in the theory of stochastic processes Markov process, for example and has been the focus of many studies Odibat and Momani, This parabolic differential equation describes the transition probability density function fx, t and is given by Oksendal In the present paper, we implemented the ADM to the time fractional F-P equation whose general form is given by.

Many definitions of fractional calculus are used to solve the problems of fractional differential equations. The most frequently encountered definitions include the Riemann-Liouville, Caputo, Wely, Rize fractional operator. We introduced the following definitions Podlubny, ; Oldham and Spanier, In this section, we consider Eq. The standard form of Eq.

Operating with I" in both sides of Eq. From this equation, the iterates are defined in the following recursive way Wazwaz, ; Adomian, ; Datta ; Adomian and Rach, Using the known f0, all components f1, f2, Substituting these fo, f1, f2, Convergence of this method has been rigorously established by Cherruault , Abbaoui and Cherruault ; and Himoun et al.

For simplicity, we take the drift term A x equal to - x and a constant fluctuation term 2k. From the relations Eq. Interestingly, as we have from Eq. Equation 3. The decomposition series solutions generally converge very rapidly in real physical problems Saha et al.

It is also worth noting that a rapid stabilization to an acceptable accuracy is evident when numerical computation of the analytic approximation is carried out Saha et al, ; Kaya, ; Ngarhasta et al, ; Shawagfeh, The advantage of this global methodology lies in the fact that it not only leads to an analytical continuous approximation which is very rapidly convergent Rida et al, ; Kaya, ; Wazwaz, ; Adomian and Rach, , but also shows the dependence, giving insight into the character and behaviour of the solution just as in a closed form solution Saha Ray et al, ; Kaya, ; Wazwaz, ; Suarez and Shokooh, The present analysis exhibits the applicability of the decomposition method to solve a fractional F-P equation.

Furthermore, this method does not require any transformation techniques, linearization, discretization of the variables and it does not make closure approximation or smallness assumptions. Definition 3. Definition 4. Definition 5. Theorem 1. Theorem 2. In , Rawashdeh and Maitama introduced and nurtured the new technique, known as natural decomposition method NDM.

In , Liao proposed homotopy analysis method HAM 47 , 48 by employing the fundamental concept of differential geometry and topology, called homotopy. The HAM has been effectively aided to find the solution for problems arises in connected areas of science and technology. In this part, we consider two examples to validate the applicability and efficiency of the proposed algorithms.

Example 1. Consider the FG equation 22 , 38 34 with initial condition Note that, and are the Adomian polynomial, which signify the nonlinear term. Example 2.

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## Fractional Analysis: Methods Of Motion Decomposition

Consider the FCH equation 22 , 26 47 with initial condition Note that, is the Adomian polynomial, which signifies the nonlinear term. In the similar way, the remaining iterative terms can be found. We can see that, as time increases, the solution of FG equation decreases. From the plots, we can say that, as time increases, the solution of FCH equation also increases. The novelty of the proposed techniques is that they provide nonlocal effect, straightforward solution procedure, promising large convergence region, moreover, free from any assumption, discretization, and perturbation.

Finally, we can conclude that both the proposed algorithms are highly methodical and more accurate, which can be used to study nonlinear problems arisen in complex phenomena. All authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

His areas of research interests are differential geometry of manifolds, applied mathematics, fractional calculus, numerical analysis, and theory of relativity. He published more than 70 research papers in national and international journal of repute. Veeresha is a research scholar in the Department of Mathematics, Karnatak University. His areas of research interests are applied mathematics, fractional calculus, numerical analysis, and differential geometry of manifolds.

His research interests include ordinary and partial differential equations, analytical methods for linear and nonlinear differential equations, mathematical physics, numerical solutions of the partial differential equations, fractional differential equations of course ordinary and partial , and computer programmings such as mathematica, Pascal, and Maple.

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