On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space

1980 ◽  
Vol 42 (3) ◽  
pp. 225-229 ◽  
Author(s):  
A. N. Leznov
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Dimensional Space ◽  
Complete Integrability ◽  
Two Dimensional ◽  
Partial Differential

scholarly journals Possibility of amoebas' aggregation in finite time

2013 ◽  
Vol 21 (1) ◽  
pp. 101-120
Author(s):  
Saïd Hilmi ◽  
Chérif Ziti
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Finite Time ◽  
Blow Up ◽  
Dimensional Space ◽  
Partial Differential ◽  
Self Similar ◽  
Similar Solutions

Abstract The dynamic of amoebas in favorable circumstances is modeled by a nonlinear system of Partial Differential Equations arising in chemotaxis. The competition between different parameters of this system plays a major role in the process of aggregation. Throughout this paper, we prove the existence of self-similar solutions that blow up in finite time in a dimensional space and under specific circumstances depending upon the position of those parameters.

scholarly journals A DIFFERENTIAL INTEGRABILITY CONDITION FOR TWO-DIMENSIONAL HAMILTONIAN SYSTEMS

2014 ◽  
Vol 54 (2) ◽  
pp. 139-141
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Partial Differential Equations ◽  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Harmonic Functions ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Coordinates

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form <em>H = T + V</em> where <em>V</em> and <em>T</em> are respectively harmonic functions of the generalized coordinates and the associated momenta.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

scholarly journals SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

2020 ◽  
Vol 5 (12) ◽  
pp. 406-420
Author(s):  
A. Aghili ◽  
M.R. Masomi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Heat Equations ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Homogeneous Systems ◽  
Fractional Pde

In this article, the authors used two dimensional Laplace transform to solve non - homogeneous sub - ballistic fractional PDE and homogeneous systems of time fractional heat equations. Constructive examples are also provided.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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.

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