A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

Bekela, Alemu Senbeta; Belachew, Melisew Tefera; Wole, Getinet Alemayehu

doi:10.1186/s13662-020-03048-3

Cited by 7 publications

(3 citation statements)

References 23 publications

(41 reference statements)

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“…Approximate analytical methods for solving some linear and nonlinear PDEs with proportional delays are considered in [163,164]. Numerical methods for solving PDEs with proportional delay are developed in [165,166], and those for PDEs with more complex varying delay are studied in [167].…”

Section: Delay Reaction-diffusion Pdesmentioning

confidence: 99%

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Polyanin

Сорокин

2023

Mathematics

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The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.

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Section: Delay Reaction-diffusion Pdesmentioning

confidence: 99%

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Polyanin

Сорокин

2023

Mathematics

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“…In [89], a finite-difference scheme for the numerical integration of firstorder PDEs with constant delay in t and proportional delay in x is constructed. Papers [90][91][92] are devoted to numerical methods for solving pantograph-type PDEs with proportional delay [90,91] and more complex varying delay [92].…”

Section: Pantograph-type Odes and Pdes And Their Applicationsmentioning

confidence: 99%

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Polyanin

Сорокин

2021

Mathematics

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We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

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“…To mention a few, we have the homotopy perturbation method (HPM) [6], the Adomian decomposition method (ADM) [7], the Laplace decomposition method (LDM) [8], the homotopy perturbation transform method (HPTM) [9], and so on. Besides using the Laplace-type integral transform [10,11], some new efficient iterative techniques with the Caputo fractional derivative [12] and Atangana-Baleanu fractional derivative [13] are developed, for example, see [14][15][16][17][18][19][20][21][22][23][24][25]. Those iterative algorithms are successfully applied to many applications in applied physical science.…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

Maitama¹,

Zhang²

2021

J. Appl. Math. Comput. Mech.

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Using the recently proposed homotopy perturbation Shehu transform method (HPSTM), we successfully construct reliable solutions of some important fractional models arising in applied physical sciences. The nonlinear terms are decomposed using He's polynomials, and the fractional derivative is calculated in the Caputo sense. Using the analytical method, we obtained the exact solution of the fractional diffusion equation, fractional wave equation and the nonlinear fractional gas dynamic equation.

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A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

Cited by 7 publications

References 23 publications

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

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