Explicit Solutions for Nonlinear Partial Differential Equations Using Bezier Functions

2007 ◽  
Author(s):  
P. Venkataraman ◽  
J. G. Michopoulos
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Metal Composite ◽  
Partial Differential ◽  
Ionic Polymer ◽  
Linear Partial Differential Equations ◽  
Polymer Metal ◽  
Set Up ◽  
Domain Discretization

This paper presents a methodology for generating solutions of non linear partial differential equations through Bezier functions. These functions define corresponding Bezier surfaces using a bipolynomial Bernstein basis function. The solution, or essentially the coefficients, is identified through design optimization. The set up is direct, elegantly simple, and involves minimizing the error in the residuals of the differential equations over the domain. No domain discretization is necessary. The procedure is not problem dependent and is adaptive through the selection of the order of the Bezier functions. Two examples: (1) the laminar flow over a flat plate; and (2) displacement of an ionic polymer-metal composite membrane are solved. Alternate solution to these problems is referenced in the paper.

Explicit Solutions for Linear Partial Differential Equations Using Bezier Functions

2006 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimization Technique ◽  
Nonlinear Boundary Value Problems ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Standard Design ◽  
Set Up ◽  
The One ◽  
Domain Discretization

Solutions in basic polynomial form are obtained for linear partial differential equations through the use of Bezier functions. The procedure is a direct extension of a similar technique employed for nonlinear boundary value problems defined by systems of ordinary differential equations. The Bezier functions define Bezier surfaces that are generated using a bipolynomial Bernstein basis function. The solution is identified through a standard design optimization technique. The set up is direct and involves minimizing the error in the residuals of the differential equations over the domain. No domain discretization is necessary. The procedure is not problem dependent and is adaptive through the selection of the order of the Bezier functions. Three examples: (1) the Poisson equation; (2) the one dimensional heat equation; and (3) the slender two-dimensional cantilever beam are solved. The Bezier solutions compare excellently with the analytical solutions.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

Analysis of Dynamic Tilt Stiffness and Damping Coefficients of Externally Pressurized Porous Gas Journal Bearings

1978 ◽  
Vol 100 (3) ◽  
pp. 359-363
Author(s):  
N. S. Rao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Theoretical Estimate ◽  
Eccentricity Ratio ◽  
Partial Differential ◽  
First Order ◽  
Linear Partial Differential Equations ◽  
Stiffness And Damping ◽  
Effect Of Feeding

A theoretical estimate of dynamic characteristics in terms of stiffness and damping of an externally pressurized gas-lubricated porous journal under tilting mode of vibration is made. The governing nonlinear partial differential equations in the porous medium and in the bearing clearance are linearized using a first-order perturbation analysis. The tilt stiffness and damping are then determined from the solution of the resulting linear partial differential equations numerically. The effect of feeding parameter, supply pressure, porosity parameter, L/D ratio and eccentricity ratio on these two above characteristics is shown.

scholarly journals On the theory of the solution of a system of simultaneous non-linear partial differential equations of the first order

1876 ◽  
Vol 24 (164-170) ◽  
pp. 337-344
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Non Linear

Given an equation of the form z = ϕ ( x 1 , x 2 , . . . x n+r , a 1 , a 2 ,... a r , a + r + 1 ), we obtain by differentiation with respect to each of the n + r variables n + r equations, together with the original equation n + r + 1 equations, from which, eliminating the r + 1 constants, we have a system of n nonlinear partial differential equations. Conversely, given a system of n non-linear partial differential equations with n + r independent variables, if there exists an equation

Exact, multiple soliton solutions of the double sine Gordon equation

1978 ◽  
Vol 359 (1699) ◽  
pp. 479-495 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Dimensional Space ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Method Of Solution

Exact, particular solutions of the double sine Gordon equation in n dimensional space are constructed. Under certain restrictions these solutions are N solitons, where N ≤ 2 q — 1 and q is the dimensionality of spacetime. The method of solution, known as the base equation technique, relates solutions of nonlinear partial differential equations to solutions of linear partial differential equations. This method is reviewed and its applicability to the double sine Gordon equation shown explicitly. The N soliton solutions have the remarkable property that they collapse to a single soliton when the wave vectors are parallel.

Correlation Moment Analysis and the Time Dependence of Coherence in Systems Described by Nonlinear Partial Differential Equations

2005 ◽  
Vol 73 (2) ◽  
pp. 197-205 ◽  
Author(s):  
M. R. Belmont
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Dependence ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Moment Analysis ◽  
Partial Differential ◽  
Linear Diffusion ◽  
Linear Partial Differential Equations ◽  
Non Linear

The work presented introduces correlation moment analysis. This technique can be employed to explore the growth of determinism from stochastic initial conditions in physical systems described by non-linear partial differential equations (PDEs) and is also applicable to wholly deterministic situations. Correlation moment analysis allows the analytic determination of the time dependence of the spatial moments of the solutions of certain types of non-linear partial differential equations. These moments provide measures of the growth of processes defined by the PDE, furthermore the results are obtained without requiring explicit solution of the PDE. The development is presented via case studies of the linear diffusion equation and the non-linear Kortweg de-Vries equation which indicate strategies for exploiting the various properties of correlation moments developed in the text. In addition, a variety of results have been developed which show how various classes of terms in PDEs affect the structure of a sequence of correlation moment equations. This allows results to be obtained about the behavior of the PDE solution, in particular how the presence of certain types of terms affects integral measures of the solution. It is also demonstrated that correlation moments provide a very simple, natural approach to determining certain subsets of conserved quantities associated with the PDEs.

Sign in / Sign up

Export Citation Format

Share Document