Implicit and explicit algorithms for solving the split feasibility problem

Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).     

On the Cauchy type problem for systems of functional differential equations

Using theorems on functional differential inequalities, we establish new efficient conditions for the solvability as well as unique solvability of the Cauchy type problem for systems of functional differential equations in both linear and nonlinear cases.     

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.     

Numerical methods based on multipoint Hermite interpolating polynomials for solving the Cauchy problem for stiff systems of ordinary differential equations

Families of A-, L-, and L(δ)-stable methods are constructed for solving the Cauchy problem for a system of ordinary differential equations (ODEs). The L(δ)-stability of a method with a parameter δ ∈ (0, 1) is defined. The methods are based on the representation of the right-hand sides of an ODE system at the step h in terms of two-or three-point Hermite interpolating polynomials. Comparative results are reported for some test problems. The multipoint Hermite interpolating polynomials are used to derive formulas for evaluating definite integrals. Error estimates are given.     

Successive column correction algorithms for solving sparse nonlinear systems of equations

This paper presents two algorithms for solving sparse nonlinear systems of equations: the CM-successive column correction algorithm and a modified CM-successive column correction algorithm. Aq-superlinear convergence theorem and anr-convergence order estimate are given for both algorithms. Some numerical results and the detailed comparisons with some previously established algorithms show that the new algorithms have some promise of being very effective in practice.This research was partially supported by contracts and grants: DOE DE-AS05-82ER1-13016, AFOSR 85-0243 at Rice University, Houston, U.S.A. and Natural Sciences and Engineering Research Council of Canada grant A-8639.     

Simulation algorithms for the second-order parabolic Cauchy problem

Monte Carlo algorithms, which solve boundary value problems for the heat equation whose elliptic part is the Laplace operator, have been known for a long time [1], [2]. They essentially use the explicit form of a fundamental solution and cannot be transferred to equations containing higher derivatives with nonconstant coefficients.     

Cauchy problem for fractional diffusion equations

We consider an evolution equation with the regularized fractional derivative of an order α∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.     

Cauchy problem for some non-Kovalewskian equations

Some evolution equations with constant leading coefficients and real characteristic roots are considered. The wellposedness of Cauchy problem is proved inH ^∞ and in Gevrey classes, if an assumption is made on the lower order terms. Finally the results are generalized to nonlinear equations.

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Sunto Si considerano equazioni differenziali di evoluzione con coefficienti costanti nella parte principale e radici caratteristiche reali. Si dimostra la buona positura del problema di Cauchy inH ^∞ e nelle classi di Gevrey, sotto un’ipotesi sui termini di ordine inferiore. Infine, i risultati vengono estesi alle equazioni non lineari.

     

The Cauchy problem for Kirchhoff equations

Conferenza tenuta il 28 settembre 1992     

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h ³) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h ³ −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h ⁵). The efficiency of the algorithms is illustrated by numerical examples.     

Cauchy problem for nonautonomous Kolmogorov equations

We study the existence and uniqueness of the strict solutions of an initial value problem for a nonautonomous Kolmogorov equation in a space of unbounded functions     

An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations

On the basis of a generalized separation-of-variables method we propose an operator method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations of first order with respect to time and of infinite order with respect to the spatial variables. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Parallel algorithms for solving retrieval equations

Translated from Vychislitel'nye Kompleksy i Modelirovanie Slozhnykh Sistem, pp. 28–36, Moscow State University, 1989.     

Stabilization of the Cauchy problem for integro-differential equations

In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1571–1576, November, 2005.     

Cauchy problem for overdetermined systems

Sunto Si caratterizzano le nozioni di direzioni caratteristiche, formalmente caratteristiche, di iperbolicità e di evoluzione in una, direzione per sistemi a coefficienti costanti, connesse allo studio delle soluzioni del problema di Cauchy in un semispazio.     

Cauchy problem for equations of internal waves

The Cauchy problem is considered for the equation of internal waves to which reduce many problems of the linear theory of waves in a continuously stratified fluid. The theorem of uniqueness is proved, and the formula for explicit representation of solution in terms of integrals whose kernels contain the obtained in /1/ fundamental solution of the internal wave operator and its time derivative are derived. Asymptotic analysis of solution in the “distant zone” is carried out for large values of dimensionless time.