scholarly journals Initial-Boundary Value Problems for Nonlinear Dispersive Equations of Higher Orders Posed on Bounded Intervals with General Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 165
Author(s):  
Nikolai A. Larkin ◽  
Jackson Luchesi
Keyword(s):  
Boundary Conditions ◽  
Present Article ◽  
Initial Boundary ◽  
Dispersive Equations ◽  
General Boundary Conditions ◽  
Initial Boundary Value Problems ◽  
Nonlinear Dispersive Equations ◽  
Mixed Problems ◽  
Decay Of Solutions ◽  
Initial Boundary Value

The present article concerns general mixed problems for nonlinear dispersive equations of any odd-orders posed on bounded intervals. The results on existence, uniqueness and exponential decay of solutions are presented.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations

2003 ◽  
Author(s):  
H. N. Narang ◽  
Rajiv K. Nekkanti
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Initial Boundary ◽  
Time Dependent ◽  
Initial Boundary Value Problems ◽  
Non Linear ◽  
Initial Boundary Value

The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.

scholarly journals Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

2011 ◽  
Vol 152 (3) ◽  
pp. 473-496 ◽  
Author(s):  
DAVID A. SMITH
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Transform Method ◽  
Arbitrary Boundary ◽  
Initial Boundary Value ◽  
Well Posed

AbstractWe study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

scholarly journals TWO INITIAL-BOUNDARY VALUE PROBLEMS WITH NONLINEAL BOUNDARY CONDITIONS FOR ONE-DIMENSION HYPERBOLIC EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 46-56
Author(s):  
L.S. Pulkina ◽  
M.V. Strigun
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Boundary Conditions ◽  
One Dimension ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

In this paper, the initial-boundary value problems for hyperbolic equationwith nonlinear boundary conditions are considered. Existence and uniqueness ofgeneralized solution are proved.

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