A (1+2)-DIMENSIONAL KELLER-SEGEL MODEL: LIE SYMMETRY AND EXACT SOLUTIONS FOR THE CAUCHY PROBLEM

BIOMAT 2014 ◽  
2015 ◽  
Author(s):  
ROMAN CHERNIHA ◽  
MAKSYM DIDOVYCH
Keyword(s):  
Cauchy Problem ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
The Cauchy Problem

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

scholarly journals On the Construction of Solutions to a Problem with a Free Boundary for the Non-linear Heat Equation

2020 ◽  
pp. 694-707
Author(s):  
Alexander L. Kazakov ◽  
Lev F. Spevak ◽  
Ming-Gong Lee
Keyword(s):  
Cauchy Problem ◽  
Free Boundary ◽  
Heat Equation ◽  
Exact Solutions ◽  
Heat Wave ◽  
Wave Type ◽  
Linear Heat Equation ◽  
Linear Heat ◽  
Non Linear ◽  
The Cauchy Problem

The construction of solutions to the problem with a free boundary for the non-linear heat equation which have the heat wave type is considered in the paper. The feature of such solutions is that the degeneration occurs on the front of the heat wave which separates the domain of positive values of the unknown function and the cold (zero) background. A numerical algorithm based on the boundary element method is proposed. Since it is difficult to prove the convergence of the algorithm due to the non-linearity of the problem and the presence of degeneracy the comparison with exact solutions is used to verify numerical results. The construction of exact solutions is reduced to integrating the Cauchy problem for ODE. A qualitative analysis of the exact solutions is carried out. Several computational experiments were performed to verify the proposed method

On exact solutions of the Cauchy problem for some parabolic and hyperbolic equations

2013 ◽  
Vol 87 (1) ◽  
pp. 12-14 ◽  
Author(s):  
V. A. Kostin ◽  
A. V. Kostin ◽  
D. V. Kostin
Keyword(s):  
Cauchy Problem ◽  
Exact Solutions ◽  
Hyperbolic Equations ◽  
The Cauchy Problem
Sign in / Sign up

Export Citation Format

Share Document