scholarly journals Lie symmetries and similarity solutions for a family of 1+1 fifth-order partial differential equations

2021 ◽  
pp. 1-16
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Partial Differential ◽  
Fifth Order

scholarly journals Lie Symmetries and Similarity Solutions for Rotating Shallow Water

2019 ◽  
Vol 74 (10) ◽  
pp. 869-877 ◽  
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shallow Water ◽  
Lie Algebra ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Polytropic Index ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Rotating Shallow Water

AbstractWe study a nonlinear system of partial differential equations that describe rotating shallow water with an arbitrary constant polytropic index γ for the fluid. In our analysis, we apply the theory of symmetries for differential equations, and we determine that the system of our study is invariant under a five-dimensional Lie algebra. The admitted Lie symmetries form the $\left\{{2{A_{1}}{\ \oplus_{s}}\ 2{A_{1}}}\right\}{\ \oplus_{s}}\ {A_{1}}$ Lie algebra for γ ≠ 1 and $2{A_{1}}{\ \oplus_{s}}\ 3{A_{1}}$ for γ = 1. The application of the Lie symmetries is performed with the derivation of the corresponding zero-order Lie invariants, which applied to reduce the system of partial differential equations into integrable systems of ordinary differential equations. For all the possible reductions, the algebraic or closed-form solutions are presented. Travel-wave and scaling solutions are also determined.

scholarly journals Similarity Solutions of Partial Differential Equations in Probability

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Mario Lefebvre
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Processes ◽  
Separation Of Variables ◽  
Similarity Solutions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Fourier Series ◽  
Dimensional Diffusion ◽  
Concentric Circles

Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

scholarly journals A NEW TECHNIQUE FOR APPROXIMATE SOLUTION OF FRACTIONAL-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
LAIQ ZADA ◽  
RASHID NAWAZ ◽  
MOHAMMAD A. ALQUDAH ◽  
KOTTAKKARAN SOOPPY NISAR
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Auxiliary Function ◽  
Partial Differential ◽  
Convergence Control ◽  
A New Technique ◽  
Fifth Order ◽  
First Time

In the present paper, the optimal auxiliary function method (OAFM) has been extended for the first time to fractional-order partial differential equations (FPDEs) with convergence analysis. To find the accuracy of the OAFM, we consider the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations as a test example. The proposed technique has auxiliary functions and convergence control parameters, which accelerate the convergence of the method. The other advantage of this method is that there is no need for a small or large parameter assumption, and it gives an approximate solution after only one iteration. Further, the obtained results have been compared with the exact solution through different graphs and tables, which shows that the proposed method is very effective and easy to implement for different FPDEs.

scholarly journals Potential symmetries of inhomogeneous nonlinear diffusion equations

2000 ◽  
Vol 61 (3) ◽  
pp. 507-521 ◽  
Author(s):  
C. Sophocleous
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Diffusion ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
Partial Differential ◽  
Potential Symmetries ◽  
Functional Forms

In this paper potential symmetries are sought of the inhomogeneous nonlinear diffusion equations ut = x1−M[xN−1f (u) ux]x. The functional forms of f (u) that admit such symmetries are completely classified. A complete list is presented of the symmetries, which depend on the values of the parameters M and N. We give examples of similarity solutions using potential symmetries. In some cases, the potential symmetries enable us to convert non–invertible mappings of nonlinear partial differential equations to linear ones.

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