Necessary and sufficient conditions for arbitrary linear combinations of solutions of a class of nonlinear partial differential equations to satisfy the differential equation

2021 ◽  
Vol 9 (1) ◽  
pp. 15-23
Author(s):  
Namik Yener
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Partial Differential ◽  
Necessary And Sufficient

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

Method of Separation of Variables for the Solution of Certain Nonlinear Partial Differential Equations

1971 ◽  
Vol 93 (2) ◽  
pp. 162-164
Author(s):  
V. A. Bapat ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Separation Of Variables ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Exact Methods ◽  
Partial Differential ◽  
Approximate Methods

A method for the solution of a certain class of nonlinear partial differential equations by the method of separation of variables is presented. The method enables the nonlinear partial differential equation to be reduced to ordinary nonlinear differential equations, which can be solved by exact methods (or by approximate methods if an exact solution is not possible).

ON SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS OF BRIOT–BOUQUET TYPE

2018 ◽  
Vol 98 (1) ◽  
pp. 122-133
Author(s):  
FENGBAI LI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Singular Solutions ◽  
Partial Differential ◽  
First Order ◽  
Associated Matrix ◽  
Necessary And Sufficient

We study systems of partial differential equations of Briot–Bouquet type. The existence of holomorphic solutions to such systems largely depends on the eigenvalues of an associated matrix. For the noninteger case, we generalise the well-known result of Gérard and Tahara [‘Holomorphic and singular solutions of nonlinear singular first order partial differential equations’, Publ. Res. Inst. Math. Sci.26 (1990), 979–1000] for Briot–Bouquet type equations to Briot–Bouquet type systems. For the integer case, we introduce a sequence of blow-up like changes of variables and give necessary and sufficient conditions for the existence of holomorphic solutions. We also give some examples to illustrate our results.

scholarly journals Transient-False Method for Solving System of Nonlinear Partial Differential Equations

2018 ◽  
Vol 1 (25) ◽  
pp. 509-522
Author(s):  
. Ali Khalaf Hussain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

          In this paper we study the false transient method  to  solve and transform a system of non-linear partial differential equations which can be solved using finite-difference method and give some problems which have a good results compared with the exact solution, whereas this method was used to transform the nonlinear partial differential equation to a linear partial differential equation which can be solved by using the alternating-direction implicit method after using the ADI method. The system of linear algebraic equations could be obtained and can be solved by using MATLAB.

scholarly journals Inextensible Flows of Spacelike Curves According to Equiform Frame in 4-dimensional Minkowski Space R41

2021 ◽  
Vol 19 ◽  
pp. 683-698
Author(s):  
W. M. Mahmoud ◽  
Alaa Hassan Noreldeen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Minkowski Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Spacelike Surface ◽  
Partial Differential ◽  
Dimensional Minkowski Space ◽  
Inextensible Flows ◽  
Necessary And Sufficient

In this paper, we study inextensible flows of spacelike curves lying fully on a spacelike surface Ω according to equiform frame in 4-dimensional Minkowski space ℝ1 4 . We give necessary and sufficient conditions for this inextensible flows which are expressed as a partial differential equation involving the equiform curvature functions in 4-dimensional Minkowski space ℝ1 4 . Finally we give an application of inextensible flows of spacelike curves in ℝ1 4 .

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