scholarly journals Exact Solutions of the Two-Dimensional Cattaneo Model Using Lie Symmetry Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Lie Symmetry ◽  
Two Dimensional ◽  
Linear Combinations ◽  
Similarity Transformations ◽  
New Exact Solutions ◽  
Symmetry Transformations

In this article, symmetry technique is utilized to obtain new exact solutions of the Cattaneo equation. The infinitesimal symmetries, linear combinations of these symmetries, and corresponding similarity variables are determined, which lead to many exact solutions of the considered equation. By applying similarity transformations, the mentioned partial differential equation is reduced to some ordinary differential equations of second order. Solutions of these ordinary differential equations have yielded many exact solutions of the Cattaneo equation.

scholarly journals Discrete Inverse Sumudu Transform Application to Whittaker Equation and Zettl Equation

2018 ◽  
Author(s):  
Rathinavel Silambarasan ◽  
Kottakkaran Sooppy Nisar ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Design Methodology ◽  
Elementary Functions ◽  
New Exact Solutions ◽  
Sumudu Transform ◽  
Sumudu Transforms

Inverse Sumudu transform multiple shifting properties are used to design methodology for solving ordinary differential equations. Then algorithm applied to solve Whittaker and Zettl equations to get their new exact solutions and profiles which shown through Maple complex graphicals. Table of inverse Sumudu transforms for elementary functions given for supporting the differential equations solving using inverse Sumudu transform.

scholarly journals New Classes of Solutions of Dynamical Problems of Plasticity

2020 ◽  
pp. 792-796
Author(s):  
Sergei I. Senashov ◽  
Olga V. Gomonova ◽  
Irina L. Savostyanova ◽  
Olga N. Cherepanova
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Science And Engineering ◽  
Spatial Variables ◽  
One Dimensional ◽  
New Exact Solutions

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

scholarly journals On Exact Solutions of Phi-4 Partial Differential Equation Using the Enhanced Modified Simple Equation Method

2018 ◽  
Vol 6 (4) ◽  
Author(s):  
Ziad Salem Rached
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Simple Equation ◽  
Equation Method ◽  
Partial Differential ◽  
New Exact Solutions ◽  
Modified Simple Equation Method

Constructing exact solutions of nonlinear ordinary and partial differential equations is an important topic in various disciplines such as Mathematics, Physics, Engineering, Biology, Astronomy, Chemistry,… since many problems and experiments can be modeled using these equations. Various methods are available in the literature to obtain explicit exact solutions. In this correspondence, the enhanced modified simple equation method (EMSEM) is applied to the Phi-4 partial differential equation. New exact solutions are obtained.

Non-Lie Symmetry Group and New Exact Solutions for the Two-Dimensional KdV-Burgers Equation

2011 ◽  
Vol 28 (2) ◽  
pp. 020205 ◽  
Author(s):  
Hong Wang ◽  
Ying-Hui Tian ◽  
Han-Lin Chen
Keyword(s):  
Exact Solutions ◽  
Symmetry Group ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Two Dimensional ◽  
New Exact Solutions

New optical soliton solutions of fractional perturbed nonlinear Schrödinger equation in nanofibers

2021 ◽  
Author(s):  
S. Saha Ray ◽  
N. Das
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
New Exact Solutions

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved [Formula: see text] expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions.

scholarly journals Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Fractional Complex Transform ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Fractional Differential

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

New exact solutions of the Boussinesq equation

1990 ◽  
Vol 1 (3) ◽  
pp. 279-300 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Boussinesq Equation ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
New Exact Solutions ◽  
New Space ◽  
Manipulation Language ◽  
Similarity Reductions

In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

scholarly journals Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Seyma Tuluce Demiray ◽  
Hasan Bulut ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Simulations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Order Derivative ◽  
Two Dimensional ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Type

We make use of the so-called Sumudu transform method (STM), a type of ordinary differential equations with both integer and noninteger order derivative. Firstly, we give the properties of STM, and then we directly apply it to fractional type ordinary differential equations, both homogeneous and inhomogeneous ones. We obtain exact solutions of fractional type ordinary differential equations, both homogeneous and inhomogeneous, by using STM. We present some numerical simulations of the obtained solutions and exhibit two-dimensional graphics by means of Mathematica tools. The method used here is highly efficient, powerful, and confidential tool in terms of finding exact solutions.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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