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.     

Monotone-iterative method for mixed boundary value problems for generalized difference equations with “maxima”

In this paper the authors investigate special type of difference equations which involve both delays and the maximum value of the unknown function over a past time interval. This type of equations is used to model a real process which present state depends significantly on its maximal value over a past time interval. An appropriate mixed boundary value problem for the given nonlinear difference equation is set up. An algorithm, namely, the monotone iterative technique is suggested to solve this problem approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima”, and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. Several numerical examples are considered to illustrate the practical application of the suggested algorithm.     

Asymptotic behavior as t→∞ of the solutions of initial-boundary value problems for the equations of motions of linear viscoelastic fluids

It is shown that for the nonstationary equations of motion of the linear viscoelastic fluids, whose defining equation has the form the stationary system is the Navier-Stokes stationary system with viscosity coefficient v: It is proved that for small Reynolds numbers the solutions of the initial-boundary value problems for the equations of motion of the Oldroyd fluids (M=L=1, 2, ...) and Kelvin-Voight fluids (M=L + 1, L=0, 1, 2, ...) converge for t to the solution of the first boundary value problem for the stationary Navier-Stokes system (*).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 171, pp. 174–181, 1989.     

Modified Tseng’s extragradient algorithms for variational inequality problems

In this work, our interest is in investigating the monotone variational inequality problems in the framework of real Hilbert spaces. For solving this problem, we introduce two modified Tseng’s extragradient methods using the inertial technique. The weak convergence theorems are established under the standard assumptions imposed on cost operators. Finally, numerical results are reported to illustrate the behavior of the new algorithms and also to compare with others.     

RBF–DQ algorithms for elliptic problems in axisymmetric domains

Numerical Algorithms - A radialbasis function (RBF)–differentialquadrature (DQ) method is applied for the numerical solution of elliptic boundary value problems (BVPs) in three-dimensional...     

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.     

A posteriori error estimates for approximations of evolutionary convection–diffusion problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.     

Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.     

Boundary value problems for an iterative di?erential equation

This paper discusses the solutions of an iterative di?erential equation under general boundary value conditions. Using an auxiliary integral equation without the help of Green’s functions usually being constructed in higher order equations, we prove the existence and uniqueness of solutions by the ?xed point theorems of Schauder and Banach, respectively. Our theorems generalize and revise the related results.     

L ∞-Estimates for nonlinear elliptic Neumann boundary value problems

In this paper we prove the L ^∞-boundedness of solutions of the quasilinear elliptic equation

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix $\bar \partial $ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.     

Gauss–Newton methods with approximate projections for solving constrained nonlinear least squares problems

This paper is concerned with algorithms for solving constrained nonlinear least squares problems. We first propose a local Gauss–Newton method with approximate projections for solving the aforementioned problems and study, by using a general majorant condition, its convergence results, including results on its rate. By combining the latter method and a nonmonotone line search strategy, we then propose a global algorithm and analyze its convergence results. Finally, some preliminary numerical experiments are reported in order to illustrate the advantages of the new schemes.