Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.; Vitanov, Kaloyan N.

doi:10.1016/j.amc.2015.07.060

Cited by 67 publications

(57 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Analytical solutions are highly appreciated, but only a few types of DE allow explicit, if any, exact analytical solutions. Recently some fractional ordinary DE and partial differential equations (PDE) were analyzed and analytically solved in [24][25][26][27][28][29][30][31][32][33][34][35][36][37]. They benefit from the use of special functions [38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Zhukovsky

2016

Axioms

View full text Add to dashboard Cite

Abstract:We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power-exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

show abstract

Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Zhukovsky

2016

Axioms

View full text Add to dashboard Cite

show abstract

“…The method of simplest equation is based on a procedure analogous to the first step of the test for the Painleve property [50] - [53]. In the version of the method called modified method of simplest equation [26] - [28] this procedure is substituted by the concept for the balance equation. We are going to discuss below the possibility of use of more than one simplest equation for obtaining exact analytical solution of the solved nonlinear partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Vitanov¹,

Dimitrova²

2018

Journal of Theoretical and Applied Mechanics

Self Cite

View full text Add to dashboard Cite

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

show abstract

“…Many approaches for obtaining exact special solutions of nonlintegrable nonlinear PDEs have been developed in the recent years, e.g., [18,24,30,38,53,70,71]. In this chapter we shall use a version of the method of simplest equation called modified method of simplest equation [30,32,33,53,56,62]. The method of simplest equation uses a procedure analogous to the first step of the test for the Painleve property [29,31,34].…”

Section: Introductionmentioning

confidence: 99%

On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation

Jordanov¹,

Vitanov²

2018

Studies in Computational Intelligence

Self Cite

View full text Add to dashboard Cite

We discuss a class of hyperbolic reaction-diffusion equations and apply the modified method of simplest equation in order to obtain an exact solution of an equation of this class (namely the equation that contains polynomial nonlinearity of fourth order). We use the equation of Bernoulli as a simplest equation and obtain traveling wave solution of a kink kind for the studied nonlinear reaction-diffusion equation.

show abstract

Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications

Cited by 67 publications

References 44 publications

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation

Contact Info

Product

Resources

About