A NOTE ON THE DIFFERENCE SCHEMES FOR HYPERBOLIC EQUATIONS

for a differential equation in a Hilbert space H with unbounded linear selfadjoint and positive definite operator A = A∗ ≥ δI (δ > 0) with dense domain D̄(A) = H . It is known (cf. [3]) that various initial boundary value problems for the hyperbolic equations can be reduced to problem (1.1). A study of discretization, over time only, of the initial value problem also permits one to include general difference schemes in applications, if the differential operator in space variables, A, is replaced by the difference operators Ah that act in the Hilbert spaces and are uniformly positive definite and selfadjoint in h for 0 < h ≤ h0. In the paper [4], the following first order accuracy difference scheme for approximately solving problem (1.1) τ−2 ( uk+1 −2uk +uk−1 )+Auk+1 = fk, fk = f (tk), tk = kτ, 1 ≤ k ≤ N −1, Nτ = 1, τ−1 ( u1 −u0 )+ iAu1 = iAu0 +ψ, u0 = φ, (1.2)


Introduction
We consider the initial value problem for a differential equation in a Hilbert space H with unbounded linear selfadjoint and positive definite operator A = A * ≥ δI (δ > 0) with dense domain D(A) = H .It is known (cf.[3]) that various initial boundary value problems for the hyperbolic equations can be reduced to problem (1.1).A study of discretization, over time only, of the initial value problem also permits one to include general difference schemes in applications, if the differential operator in space variables, A, is replaced by the difference operators A h that act in the Hilbert spaces and are uniformly positive definite and selfadjoint in h for 0 < h ≤ h 0 .In the paper [4], the following first order accuracy difference scheme for approximately solving problem (1.1) was considered.The stability estimates for the solution of the difference scheme (1.2) were obtained.The proof of these statements is based on the transform of second order difference equations to equivalent system of first order difference equations.Application of this approach in [1,2] with similar results for the solutions of the second order accuracy of the following difference schemes for approximately solving the initial value problem (1.1) were obtained.However, for practical realization of these difference schemes it is necessary to first construct an operator A 1/2 .This action is very difficult for a realization.Therefore, in spite of theoretical results the role of their application to a numerical solution for an initial value problem is not great.
In the present paper, first and second order accuracy difference schemes for approximate solutions of problem (1.1) are constructed using the integer powers of the operator A, and the stability estimates for the solution of these difference schemes are obtained.

First order difference schemes
We consider the first order accuracy difference scheme for approximately solving the initial value problem (1.1) (2. 2) The proof of this theorem uses the method of [4] and is based on the following formulas: ) −1 and on the estimates (2.4) Note that formulas (2.3) are generated by the operator A 1/2 and are used to prove stability estimates for the solutions of the difference scheme (2.1).
However, for the practical realization of this difference scheme (2.1) the operator A 1/2 as in [1,2,4] is not used.Note also that these stability inequalities in the case k = 1 are weaker than the respective inequalities in the cases k = 2, . . ., N. However, obtaining this type of inequalities is important for applications.We denote by a τ = (a k ) the mesh function of approximation.Then if we assume that τ Aa 1 H tends to 0 as τ tends to 0 not slower than a 1 H .It takes place in applications by supplementary restriction on the smoothness property of the data in the space variables.
It is clear that the estimate is absent.However, estimates for the solution of first order accuracy modification difference scheme for approximately solving the initial value problem (1.1) are better than the estimates for the solution of the difference scheme (2.1).
Then for the solution of the difference scheme (2.6), the following stability inequalities, for 1 ≤ k ≤ N, hold (2.7) The proof of this theorem is based on the following formulas: A. Ashyralyev and P. E. Sobolevskii 67 and on the estimates (2.4).

Second order difference schemes
We consider the second order accuracy difference schemes for approximate solutions of the initial value problem (1.1) The stability estimates for the solution of these difference schemes are obtained.
Then for the solution of the difference scheme (3.1), the following stability inequalities, for The proof of this theorem is based on the following formulas: where (3.5) A. Ashyralyev and P. E. Sobolevskii 69 Then for the solution of the difference scheme (3.2), the following stability inequalities, for 1 ≤ k ≤ N, hold The proof of this theorem is based on the following formulas: 70 A note on the difference schemes for hyperbolic equations where and on the estimates (3.8)

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