scholarly journals Closed-Form Solutions for a Nonlinear Partial Differential Equation Arising in the Study of a Fourth Grade Fluid Model

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Taha Aziz ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Closed Form ◽  
Fluid Model ◽  
Nonlinear Partial Differential Equation ◽  
Fourth Grade ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Grade Fluid ◽  
Fourth Grade Fluid

scholarly journals Shock Wave Solution for a Nonlinear Partial Differential Equation Arising in the Study of a Non-Newtonian Fourth Grade Fluid Model

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Taha Aziz ◽  
A. Fatima ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Shock Wave ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Fluid Model ◽  
Nonlinear Partial Differential Equation ◽  
Fourth Grade ◽  
Shock Wave Solution ◽  
Grade Fluid ◽  
Fourth Grade Fluid

This study focuses on obtaining a new class of closed-form shock wave solution also known as soliton solution for a nonlinear partial differential equation which governs the unsteady magnetohydrodynamics (MHD) flow of an incompressible fourth grade fluid model. The travelling wave symmetry formulation of the model leads to a shock wave solution of the problem. The restriction on the physical parameters of the flow problem also falls out naturally in the course of derivation of the solution.

Group Theoretical Analysis and Invariant Solutions for Unsteady Flow of a Fourth-Grade Fluid over an Infinite Plate Undergoing Impulsive Motion in a Darcy Porous Medium

2015 ◽  
Vol 70 (7) ◽  
pp. 483-497 ◽  
Author(s):  
Taha Aziz ◽  
Aeeman Fatima ◽  
Asim Aziz ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Group ◽  
Group Analysis ◽  
Fourth Grade ◽  
Invariant Solutions ◽  
Conditional Symmetry ◽  
Partial Differential ◽  
Grade Fluid ◽  
Fourth Grade Fluid

AbstractIn this study, an incompressible time-dependent flow of a fourth-grade fluid in a porous half space is investigated. The flow is generated due to the motion of the flat rigid plate in its own plane with an impulsive velocity. The partial differential equation governing the motion is reduced to ordinary differential equations by means of the Lie group theoretic analysis. A complete group analysis is performed for the governing nonlinear partial differential equation to deduce all possible Lie point symmetries. One-dimensional optimal systems of subalgebras are also obtained, which give all possibilities for classifying meaningful solutions in using the Lie group analysis. The conditional symmetry approach is also utilised to solve the governing model. Various new classes of group-invariant solutions are developed for the model problem. Travelling wave solutions, steady-state solution, and conditional symmetry solutions are obtained as closed-form exponential functions. The influence of pertinent parameters on the fluid motion is graphically underlined and discussed.

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

EXACT SOLUTIONS FOR INTERFACIAL OUTFLOWS WITH STRAINING

2014 ◽  
Vol 55 (3) ◽  
pp. 232-244 ◽  
Author(s):  
LAWRENCE K. FORBES ◽  
MICHAEL A. BRIDESON
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Point Source ◽  
Exact Solutions ◽  
Closed Form ◽  
Line Source ◽  
Nonlinear Partial Differential Equation ◽  
Background Flow ◽  
Closed Form Solutions ◽  
Fluid Instabilities

AbstractIn models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

scholarly journals Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 24
Author(s):  
Muhammad Shakeel ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Fractional Order ◽  
Soliton Solutions ◽  
Wave Transformation ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Closed Form Solutions

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

scholarly journals EVOLUTION OF MODULATIONAL INSTABILITY IN TRAVELLING WAVE SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

2020 ◽  
Vol 5 (1) ◽  
pp. 1-7
Author(s):  
Ram Dayal Pankaj ◽  
Arun Kumar ◽  
Chandrawati Sindhi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Modulational Instability ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Partial Differential ◽  
Jacobian Elliptic Functions

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters

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