Closed form solutions to the coupled space-time fractional evolution equations in mathematical physics through analytical method

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

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Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.5633 ◽

2019 ◽

Vol 42 (18) ◽

pp. 5953-5968 ◽

Cited By ~ 3

Author(s):

Emad A.‐B. Abdel‐Salam ◽

Mohamed F. Mourad

Keyword(s):

Evolution Equations ◽

Space Time ◽

Fractional Evolution Equations

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Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

Modern Physics Letters B ◽

10.1142/s0217984921502523 ◽

2021 ◽

pp. 2150252

Author(s):

Sachin Kumar ◽

Monika Niwas

Keyword(s):

Exact Solutions ◽

Closed Form ◽

Evolution Equations ◽

Dynamical Behavior ◽

Lie Symmetry ◽

Nonlinear Evolution Equations ◽

Mathematical Methods ◽

Closed Form Solutions ◽

Symmetry Reductions ◽

Engineering Sciences

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.

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Closed Form Solutions to Nonlinear Heat Conduction Problems

Defect and Diffusion Forum ◽

10.4028/www.scientific.net/ddf.326-328.115 ◽

2012 ◽

Vol 326-328 ◽

pp. 115-119

Author(s):

J. Zabadal ◽

R. Garcia ◽

V. Ribeiro ◽

F. Van Der Laan

Keyword(s):

Heat Conduction ◽

Partial Differential Equations ◽

Closed Form ◽

Control Systems ◽

Analytical Method ◽

Linear Models ◽

Nonlinear Partial Differential Equations ◽

Nonlinear Heat Conduction ◽

Closed Form Solutions ◽

Standard Techniques

This work presents a new analytical method for solving nonlinear heat conduction problems in arbitrary domains. The method is based on approximate mappings which transforms nonlinear partial differential equations into linear models which can be solved using standard techniques. In order to verify whether the proposed formulation can be employed to conceive new online control systems, numerical results are reported.

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