White noise theory and general improved Kudryashov method for stochastic nonlinear evolution equations with conformable derivatives

Abd-Allah Hyder

doi:10.1186/s13662-020-02698-7

Abd-Allah Hyder

Open Access

https://doi.org/10.1186/s13662-020-02698-7

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Abstract

The aim of this work is to investigate the Wick-type stochastic nonlinear evolution equations with conformable derivatives. The general Kudryashov method is improved by a new auxiliary equation. So, a new technique, which we call “the general improved Kudryashov method (GIKM)”, is introduced to produce exact solutions for the nonlinear evolution equations with conformable derivatives. By means of GIKM, white noise theory, Hermite transform, and computerized symbolic computation, a novel technique is presented to solve the Wick-type stochastic nonlinear evolution equations with conformable derivatives. This technique is applied to construct exact traveling wave solutions for Wick-type stochastic combined KdV–mKdV equation with conformable derivatives. Moreover, numerical simulations with 3D profiles are shown for the obtained results.

Highlights

Nonlinear evolution equations have a significant job in applied sciences, especially in physics
The major feature of the general improved Kudryashov method (GIKM) over the others lies in the way that it utilizes an especially clear and powerful algorithm to obtain exact solutions for large families of nonlinear evolution equations
To construct exact solutions of the random combined Korteweg–de Vries (KdV)–Modified Korteweg–de Vries (mKdV) equation, we only examine it in a white noise ambience, we will investigate the Wick-type stochastic combined KdV– mKdV equation (26)

Summary

Introduction

Nonlinear evolution equations have a significant job in applied sciences, especially in physics. On account of [14, 33,34,35,36,37,38,39], many kinds of Wick-type stochastic and fractional evolution equations have been studied by utilizing diverse expansion techniques and white noise analysis. A modified Kudryashov manner was presented by exchanging the ordinary exponential function eκ by means of the general sort of the exponential function aκ in [41,42,43,44] In these contributions, experts got the exact solutions to the NPDE (1) by using the expansion u(κ) =. We apply the GIKM and white noise topics to construct exact solutions for the Wick-type stochastic combined KdV–mKdV equation with conformable derivatives. 4, we apply the GIKM, jointly with the Gaussian analysis of white noise, to investigate the Wick-type stochastic combined KdV–mKdV equation with conformable derivatives.

Basilar topics of white noise discipline

Demonstration of the GIKM Consider a conformable NPDE in the form

Conclusion