A robust and accurate solver for some nonlinear partial differential equations and tow applications

Abstract

In this paper we introduce a robust and accurate solver in order to solve various classes of nonlinear partial differential equations. This solver gives the closed formula for the solutions. Within this respect, a solver is developed to accurately resolve and represent the complete wave structure of the nonlinear partial differential equations. To verify this solver, the two applications are introduced. The obtained solutions may be applicable for some new observations in physics, fluid mechanics, biology, engineering, economy, astrophysics, meteorological topics, gravitational collapse and laser technology. The theoretical study and obtained solutions reported illustrating that the proposed solver efficient and adequate. The shape for the obtained solutions are illustrated via 2D and 3D plots for suitable values of the free parameters.

Exact solutions of the cubic Boussinesq and the coupled Higgs system

“…Non-linear evolution equations (NLEE) are significant type of PDE having applications in many different branches of science and engineering including quantum mechanics, optical fibres, relativity, plasma, nuclear industry, heat flow, biology, statistical mechanics etc., [1][2][3][4][5][6][7][8][9][10][11]. Different types of traveling wave solutions including exponential, rational, hyperbolic, trigonometric, dark, bright, complex, elliptic, and Jacobi elliptic, functions model many phenomena in science.…”

The new structures of stochastic solutions for the nonlinear Schrödinger’s equations

“…In view of the unified solver technique introduced in ref. 21, the solutions for equation (2.1) are Rational solutions: (at C 3 = 0)…”

“…In ref. 21, we introduced a unified solver technique to solve equation (1.1) in deterministic case in a completely unified way. The effect of stochastic terms of the NPDEs is so important to explain many vital phenomena in many fields of real life problems, such as fluid mechanics, biology, engineering, chemical engineering, fluid dynamics, solid state physics, signal processing.…”

Stochastic solutions to the non-linear Schrodinger equation in optical fiber

“…In view of the unified solver technique introduced in [19], the random behaviour for eq. ( 1): -Rational function solutions…”