On Exact Solutions of Linear PDEs

2003 ◽  
pp. 13-29 ◽  
Author(s):  
Richard Beals
Keyword(s):  
Exact Solutions ◽  
Linear Pdes

scholarly journals Integrability of Differential Equations with Fluid Mechanics Application: from Painleve Property to the Method of Simplest Equation

2013 ◽  
Vol 43 (2) ◽  
pp. 31-42 ◽  
Author(s):  
Zlatinka I. Dimitrova ◽  
Kaloyan N. Vitanov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Exact Solutions ◽  
Inverse Scattering ◽  
Nonlinear Pde ◽  
Partial Differential ◽  
Modified Method ◽  
Painlevé Property ◽  
Linear Pdes

Abstract We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order possesses the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani- Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by ex- act reduction of this PDE possesses the Painleve property. The Painleve property motivated much research on obtaining exact solutions on non- linear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.

Lump and interaction solutions to linear PDEs in 2+1 dimensions via symbolic computation

2019 ◽  
Vol 33 (36) ◽  
pp. 1950457 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Symbolic Computation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Linear Partial Differential Equations ◽  
Linear Pdes

The aim of this paper is to show that there exist lump solutions and interaction solutions to linear partial differential equations in 2[Formula: see text]+[Formula: see text]1 dimensions. Through symbolic computations with Maple, we exhibit a great variety of exact solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional linear partial differential equations, and present a specific example which possesses lump, lump-kink and lump-soliton solutions. This supplements the study on lump, rogue wave and breather solutions and their interaction solutions to nonlinear integrable equations.

scholarly journals A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

2022 ◽  
Vol 30 (1) ◽  
pp. 335-361
Author(s):  
Melih Cinar ◽  
◽  
Ismail Onder ◽  
Aydin Secer ◽  
Mustafa Bayram ◽  
...  
Keyword(s):  
Dynamical System ◽  
Differential Operator ◽  
Exact Solutions ◽  
3D Graphics ◽  
Langmuir Waves ◽  
New Exact Solutions ◽  
Different Types ◽  
Pde Systems ◽  
2D And 3D ◽  
Linear Pdes

<abstract><p>This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $ M- $ truncated derivatives. The solutions are compared in the $ 2D $ and $ 3D $ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.</p></abstract>

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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