scholarly journals REGULARITY THEOREMS FOR SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

1957 ◽  
Vol 43 (2) ◽  
pp. 234-236 ◽  
Author(s):  
F. E. Browder
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Partial Differential

The Invariant Subspace Method for Solving Fractional Partial Differential Equations

2020 ◽  
pp. 267-282
Author(s):  
Mohamed Soror Abdel Latif ◽  
Abass Hassan Abdel Kader
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Balance Principle ◽  
New Exact Solutions

In this chapter, the authors discuss the effectiveness of the invariant subspace method (ISM) for solving fractional partial differential equations. For this purpose, they have chosen a nonlinear time fractional partial differential equation (PDE) with variable coefficients to be investigated through this method. One-, two-, and three-dimensional invariant subspace classifications have been performed for this equation. Some new exact solutions have been obtained using the ISM. Also, the authors give a comparison between this method and the homogeneous balance principle (HBP).

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.

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