scholarly journals On the peakon inverse problem for the Degasperis-Procesi equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Keivan Mohajer
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Real Line ◽  
Boundary Value ◽  
The Real ◽  
String Type ◽  
Biorthogonal Polynomials ◽  
Type Boundary

Abstract The peakon inverse problem for the Degasperis-Procesi equation is solved directly on the real line, using Cauchy biorthogonal polynomials, without any additional transformation to a “string”-type boundary value problem known from prior works.

scholarly journals On the solvability of a boundary value problem on the real line

2011 ◽  
Vol 2011 (1) ◽  
pp. 26 ◽  
Author(s):  
Giovanni Cupini ◽  
Cristina Marcelli ◽  
Francesca Papalini
Keyword(s):  
Boundary Value Problem ◽  
Real Line ◽  
Boundary Value ◽  
The Real

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

Method of Forced Monotonicity for Conjugate type Boundary Value Problems for Ordinary Differential Equations

1988 ◽  
Vol 31 (1) ◽  
pp. 79-84
Author(s):  
P. W. Eloe ◽  
P. L. Saintignon
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Linear Differential ◽  
Type Boundary ◽  
Sign Conditions ◽  
Conjugate Type

AbstractLet I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

Three solutions of an th order three-point focal type boundary value problem

2008 ◽  
Vol 69 (10) ◽  
pp. 3386-3404 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Patricia J.Y. Wong ◽  
Bo Yang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Solutions ◽  
Type Boundary

Approximate Analytical Solutions to Nonlinear Inverse Boundary Value Problems

2016 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Analytical Solutions ◽  
Traditional Approach ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
Approximate Analytical Solutions ◽  
Inverse Boundary Value Problems

A nontraditional approach to the nonlinear inverse boundary value problem is illustrated using multiple examples of the Poisson equation. The solutions belong to a class of analytical solutions defined through Bézier functions. The solution represents a smooth function of high order over the domain. The same procedure can be applied to both the forward and the inverse problem. The solution is obtained as a local minimum of the residuals of the differential equations over many points in the domain. The Dirichlet and Neumann boundary conditions can be incorporated directly into the function definition. The primary disadvantage of the process is that it generates continuous solution even if continuity and smoothness are not expected for the solution. In this case they will generate an approximate analytical solution to either the forward or the inverse problem. On the other hand, the method does not need transformation or regularization, and is simple to apply. The solution is also good at damping the perturbations in measured data driving the inverse problem. In this paper we show that the method is quite robust for linear and nonlinear inverse boundary value problem. We compare the results with a solution to a nonlinear inverse boundary value problem obtained using a traditional approach. The application involves a mixture of symbolic and numeric computations and uses a standard unconstrained numerical optimizer.

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