This comparative study of fractional nonlinear fractional Burger’s equations and their systems has been done using two efficient analytical techniques. The generalized schemes of the proposed techniques for the suggested problems are obtained in a very sophisticated manner. The numerical examples of Burger’s equations and their systems have been solved using Laplace residual power series method and Elzaki transform decomposition method. The obtained results are compared through graphs and tables. The error tables have been constructed to show the associated accuracy of each method. The procedures of both techniques are simple and attractive and, therefore, can be extended to solve other important fractional order problems.

<abstract><p>In the current note, we broaden the utilization of a new and efficient analytical computational scheme, approximate analytical method for obtaining the solutions of fractional-order Fokker-Planck equations. The approximate solution is obtained by decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. The Caputo-Riemann operator property for fractional-order partial differential equations is calculated through the utilization of the provided initial source. This analytical scheme generates the series form solution which is fast convergent to the exact solutions. The obtained results have shown that the new technique for analytical solutions is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.</p></abstract>

In this article, a new modification of the Adomian decomposition method is performed for the solution fractional order convection–diffusion equation with variable coefficient and initial–boundary conditions. The solutions of the suggested problems are calculated for both fractional and integer orders of the problems. The series of solutions of the problems with variable coefficients have been provided for the first time. To verify and illustrate our new technique, four numerical examples are presented and solved by using the proposed technique. The derived results are plotted, and the dynamics are shown for both fractional and integer orders of the problems. An excellent variation among the solutions at various fractional orders is observed. It is analyzed that the new technique based on the Adomian decomposition method is accurate and effective. The present method fits both the initial and boundary conditions with double approximations simultaneously, which increases the accuracy of the present method. For the first time, the present technique is used for the solutions of the problems with variable coefficients along with initial and boundary conditions. It is therefore suggested to apply the present procedure for the solutions of other problems with variable order and coefficients along with initial and boundary conditions.

In this article, the solutions of higher nonlinear partial differential equations (PDEs) with the Caputo operator are presented. The fractional PDEs are modern tools to model various phenomena more accurately. The residual power series method (RPSM) is used for the solution analysis of fractional partial differential equations (FPDEs), which has direct implementation for the solutions of fractional partial differential equations. In this work, the solutions to a few nonlinear FPDEs are handled by the proposed technique. The general and particular schemes of RPSM are constructed and implemented successfully. The fractional solutions of PDEs have provided many useful dynamics of the targeted problems. The RPSM results for both integer and fractional-order FPDEs are further explained and elaborated by using graphs and tables. It is observed that the higher accuracy of RPSM is achieved with fewer calculations. Graphs and tables for fractional-order solutions are presented, which confirm the convergence phenomena of fractional solutions toward integer order solutions of each problem. The suggested method can be extended to the solutions of other nonlinear fractional partial differential equations.

The solutions to fractional differentials equations are very difficult to investigate. In particular, the solutions of fractional partial differential equations are challenging tasks for mathematicians. In the present article, an extension to this idea is presented to obtain the solutions of non-linear fractional Korteweg–de Vries equations. The solutions comparison of the proposed problems is done via two analytical procedures, which are known as the Residual power series method (RPSM) and q-HATM, respectively. The graphical and tabular analysis are presented to show the reliability and competency of the suggested techniques. The comparison has shown the greater contact between exact, RPSM, and q-HATM solutions. The fractional solutions are in good control and provide many important dynamics of the given problems.