Solving inverse problems for variational equations using “generalized collage methods,” with applications to boundary value problems

A number of inverse problems involving deterministic and random differential equations may be viewed in terms of the problem of approximating a target element $x$ of a complete metric space $(X,d)$ by the fixed point $\stackrel{?}{x}$ of a contraction mapping $T:X\to X$. Most practical methods rely on a reformulation of this problem due to the “Collage Theorem,” a simple consequence of Banach’s Fixed Point Theorem: They search for a contraction mapping that minimizes the “collage distance” $d(x,Tx)$. One may consider the collage method as a kind of regularization procedure for the inverse problem. In this paper, after recalling some applications of the Collage Theorem to the solution of inverse problems for fixed point equations and applications of it to initial value problems, with the help of the Lax–Milgram representation, we develop some generalizations of the collage method in order to solve inverse problems for variational equations. We consider both deterministic and stochastic problems. We then show some applications to inverse boundary value problems.     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

Initial–boundary value problems for continuity equations with BV coefficients

We establish well-posedness of initial–boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless of the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.     

Nonhomogeneous boundary value problems for some nonlinear equations with singular ?-Laplacian

Using Leray-Schauder degree theory we obtain various existence results for the quasilinear equation problems

(?(u^′)^)′=f(t,u,u^′)     

L ∞ andW 1, ∞ estimates for second order generalized difference methods on two-point boundary value problems

In this paper, we consider the second order generalized difference scheme for the two-point boundary value problem and obtain optimal order error estimates inL ^∞ andW ^{1, ∞}. The results in this paper perfect the theory of the second order generalized difference method.     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

A generalized Laguerre–Legendre spectral collocation method for solving initial-boundary value problems

In this research, a mixed spectral collocation method based on Kronecker product is proposed for solving initial-boundary value problems. New implementation is suggested to achieve more accurate approximation at longer times. Test problems are also studied to demonstrate how this method is implemented. Numerical experiments reveal that the new method is very effective and convenient.     

On boundary value problems for the ϕ-Laplacian

For the φ-Laplacian, we consider a boundary value problem with functional boundary conditions. The Dirichlet problem is a special case of this problem.     

Symmetric positive solutions for nonlinear boundary value problems with ϕ-Laplacian operator

This paper proves the existence, multiplicity, and nonexistence of symmetric positive solutions to nonlinear boundary value problems with Laplacian operator. We improve and generalize some relative results. Our analysis mainly relies on the fixed point theorem of cone expansion and compression of norm type.     

Asymptotic behavior of solutions for initial–boundary value problems for strongly damped nonlinear wave equations

We study initial–boundary value problems for strongly damped nonlinear wave equations. By using improved integral estimates, it is proven that the solutions of the problems decay to zero exponentially as time t$t$ approaches infinity, under a very simple and general assumption regarding the nonlinear term.     

A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.     

On the solvability of initial boundary value problems for nonlinear time-dependent Schrödinger equations

In this paper, we study the initial boundary value problems for a nonlinear time-dependent Schrödinger equation with Dirichlet and Neumann boundary conditions, respectively. We prove the existence and uniqueness of solutions of the initial boundary value problems by using Galerkin’s method.