A NOTE ON THE DIFFERENCE SCHEMES FOR HYPERBOLIC-ELLIPTIC EQUATIONS

The nonlocal boundary value problem for hyperbolic-elliptic equation d2u(t)/dt2 +Au(t) = f (t), (0≤ t ≤ 1), −d2u(t)/dt2 +Au(t) = g(t), (−1≤ t ≤ 0), u(0)= φ, u(1)= u(−1) in a Hilbert space H is considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.


Introduction
It is known (see [14,15,19,20]) that various boundary value problems for the hyperbolicelliptic equations can be reduced to the nonlocal boundary value problem for differential equation in a Hilbert space H, with the self-adjoint positive definite operator A.
A function u(t) is called a solution of problem (1.1) if the following conditions are satisfied.
(i) u(t) is twice continuously differentiable in the region [−1,0) (0,1] and continuously differentiable on the segment [−1 ,1].The derivative at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii) The element u(t) belongs to D(A) for all t ∈ [−1,1], and the function Au(t) is continuous on [−1,1].(iii) u(t) satisfies the equation and boundary value conditions (1.1).
In the paper [13] the first order of accuracy difference scheme for approximately solving the boundary value problem (1.1) Theorem 1.2 [6].Let ϕ ∈ D(A).Then for the solution of the difference scheme (1.3) obey the stability inequalities where M does not depend on τ, ϕ, and Methods for numerical solutions of the nonlocal boundary value problems for partial differential equations have been studied extensively by many researches (see [1, 2, 5, 3, 4, 7-9, 11, 12, 16-18, 21, 22] and the references therein).
In present paper the second order of accuracy difference schemes approximately solving the boundary-value problem (1.1) are presented.The stability estimates for the solution of these difference schemes are established.

The second order of accuracy difference schemes
Applying the second order of accuracy difference schemes of paper [10] for hyperbolic equations and the second order of accuracy difference scheme for elliptic equations we will construct the following second order of accuracy difference schemes for approximately solving the boundary value problem (1.1): 4 Difference schemes for hyperbolic-elliptic equations Theorem 2.1.Let ϕ ∈ D(A).Then for the solution of the difference scheme (2.1) obey the stability inequalities where M does not depend on τ, ϕ, and The proof of Theorem 2.1 follows the scheme of the proof of Theorem 1.2 is based on the formulas and on the estimates and on the following lemmas.
A. Ashyralyev et al. 7 Proof.Since (2.15) (2.16) to prove (2.14) it suffices to establish the estimate ( Here (2.19) The estimate (2.17) was proved in [19].Finally, using the identity where M does not depend on τ, ϕ, and The proof of Theorem 2.4 follows the scheme of the proof of Theorem 1.2 is based on the formulas 10 Difference schemes for hyperbolic-elliptic equations and on the estimates (2.6) and and on the following lemmas.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation