Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

Kumam

et al. 2019

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Kumam

et al. 2019

“…Entropies based on fractional calculus [ 7 ], integer and fractional dynamical systems can be solved by entropy analysis [ 8 ], nonlinear partial differential equations [ 9 ] in entropy and convexity, as well as ractional derivative advection–diffusion in two-dimensional semi-conductor systems and the dynamics of a national soccer league [ 10 , 11 ]. The exact solution to differential equations (DEs) of fractional order with mixed partial derivatives [ 12 ] are fractional linear differential equations with constant coefficients in Laplace transform [ 13 ]. Laplace homotopy analysis method (LHAM) can be used to solve FDEs [ 14 ] systems of non-linear FPDEs in a new analytical technique [ 15 ].…”

Section: Introductionmentioning

confidence: 99%

Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Arif

et al. 2019

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

“…Feng’s first integral method was applied successfully to obtain nonlinear space-time fractional modified Korteweg–de Vries equations [ 7 ], nonlinear partial differential equations, third-order dispersion [ 8 , 9 ] in entropy and convexity, fractional derivative advection-diffusion in two-dimensional semi-conductor systems, and the dynamics of a national soccer league [ 10 ]. The exact solution to differential equations (DEs) of fractional-order with mixed partial derivatives [ 11 ] and space-fractional diffusion equation and Tsallis relative entropy [ 12 ]. Diffusion forms contrast from regular diffusion in that the scattering of particles continues quicker (super diffusion) or slower (sub diffusion) than for the ordinary case.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method

Mustafa

et al. 2019

In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.