scholarly journals The massive Thirring system in the quarter plane

2019 ◽  
Vol 150 (5) ◽  
pp. 2387-2416
Author(s):  
Baoqiang Xia
Keyword(s):  
Integrable Model ◽  
Initial Boundary ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Transform Method ◽  
Scattering Transform ◽  
Quarter Plane ◽  
Schwartz Class ◽  
Fokas Method ◽  
Initial Boundary Value

AbstractThe unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

scholarly journals Fokas method for linear boundary value problems involving mixed spatial derivatives

2020 ◽  
Vol 476 (2239) ◽  
pp. 20200076
Author(s):  
A. Batal ◽  
A. S. Fokas ◽  
T. Özsarı
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Parabolic Pdes ◽  
Linear Change ◽  
Transform Method ◽  
Change Of Variables ◽  
Mixed Derivatives ◽  
Fokas Method ◽  
Initial Boundary Value

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

scholarly journals Evolution equations on time-dependent intervals

2019 ◽  
Vol 84 (5) ◽  
pp. 1044-1060
Author(s):  
Athanasios S Fokas ◽  
Beatrice Pelloni ◽  
Baoqiang Xia
Keyword(s):  
Evolution Equations ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Time Dependent ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Linear Evolution ◽  
First Case ◽  
Linear Integral ◽  
Initial Boundary Value

Abstract We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

scholarly journals Unified Transform method for the Schr¨odinger Equation on a Simple Metric Graph

2019 ◽  
pp. 412-420
Author(s):  
Gulmirza Khudayberganov ◽  
Zarifboy A. Sobirov ◽  
Mardonbek R. Eshimbetov
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Metric Graphs ◽  
Transform Method ◽  
Star Graphs ◽  
Representation Of Solutions ◽  
Branching Point ◽  
Fokas Method ◽  
The Fourier Transform ◽  
Initial Boundary Value

Integral-representation of solutions of the initial-boundary value problems for the Schr¨odinger equation on simple metric graphs was obtained with the use of the Fokas method. This method uses special gen- eralization of the Fourier transform that is referred to as the unified transform. Obtained representation of solutions of the problem for open and closed simple star graphs allows one to identify transmitted, reflected and trapped waves at the graph branching point

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boundary Values ◽  
Internal Resonances ◽  
Specific Parameter ◽  
Plate Equations ◽  
Initial Boundary Value ◽  
Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

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 Homotopy Perturbation Transform Method for Extensible Beam Equations

2019 ◽  
Vol 8 (1) ◽  
pp. 15-20
Author(s):  
Khaled A. Ishag ◽  
Faris Azhari Okasha
Keyword(s):  
Nonlinear Equation ◽  
Analytical Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fixed Distance ◽  
Beam Equations ◽  
Initial Boundary Value ◽  
Transverse Deflection

In this paper, we apply analytical method (homotopy perturbation transformmethod), for solving extensible beam, we discuss certain initial- boundary value problems for the nonlinear equation. This equation wasproposed by Woiniwsky- Krieger as a model for transverse deflection of an extensible beam of natural length whose ends are held a fixed distance apart.

Stochastic diffusion equation with fractional Laplacian on the first quadrant

2019 ◽  
Vol 22 (3) ◽  
pp. 795-806
Author(s):  
Jorge Sanchez-Ortiz ◽  
Francisco J. Ariza-Hernandez ◽  
Martin P. Arciga-Alejandre ◽  
Eduard A. Garcia-Murcia
Keyword(s):  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stochastic Evolution ◽  
Stochastic Diffusion Equation ◽  
Representation Of Solutions ◽  
Fokas Method ◽  
Initial Boundary Value ◽  
Main Ideas

Abstract In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

Sign in / Sign up

Export Citation Format

Share Document