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  • 1
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Petrov-Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H1 Galerkin can be interpreted within the Petrov-Galerkin projection method framework.The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov-Galerkin numerical process are proposed.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let A be a selfadjoint uniformly elliptic differential operator. Let the underlying domain be bounded. Eigenvalue problems can be solved then, and an arbitrary square integrable function may be developed in a Fourier series relative to the eigenfunctions. In general elliptic differential operators have a continuous spectrum, if the underlying domain is unbounded. In this case the spectral theorem provides a representation of a given function by an integral transformation. The spectral projector can be calculated, if the outgoing and incoming solutions are known (radiation condition). Thus integral transformations may be derived very easily. Four examples will be given: the Fourier sine transform, the Lebedev transform, the transformation belonging to the Dirichlet problem of the plate equation and, finally, the Fourier transformation.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new proof of the sampling theorem for non-bandlimited signals is presented. It leads to better rates of convergence for the envolved approximation process. Furthermore, the order of convergence is given for an extended class of functions.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Pulse mode sonar operation is analyzed under the assumption that the scattering object Γ lies in the far fields of both the transmitter and the receiver. It is shown that, in this approximation, the sonar signal is a plane wave s(x · θ0 - t) near Γ, where θ0 is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e(x · θ - t) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, if Γ is stationary with respect to the sonar system then it is shown that where š(ω) is the Fourier transform of S(τ) and T+(ωθ,ωθ0) is the scattering amplitude in the direction θ due to the scattering by Γ of a time-harmonic plane wave with frequence ω and propagation direction θ0. A generalization of this relation is derived for moving scatterers.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The interior initial-boundary value problem for Maxwell's equations is considered for certain homogeneous boundary conditions and currents which satisfy corresponding compatibility conditions. Under the assumption that a solution exists, it is shown by means of suitable normed spaces that the sequence of discrete solutions obtained from the horizontal line method converges discretely to this solution. A priori estimations are given both for the discrete solutions and the solution of the given problem.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions φλ to the equation φλ - K λ φ λ = fλ as λ → 0 where {Kλ: X → X, λ ∊ (0,α)}, α 〉 0, denotes a family of compact linear operators such that I - Kλ (I identity) is bijective for λ∊(0,α) whilst I - K0 is not and ‖ Kλ - K0‖ →, 0, ‖fλ - f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider a simple model of a jet engine inlet consisting of two coaxial cylinders. The inner cylinder extends in both directions to infinity and represents the hub of a jet engine. The outer cylinder is assumed to be semiinfinite representing the engine's jacket. This arrangement is approached by a compressible and invisced gas moving at a velocity U less than the speed of sound. Two problems are formulated. In problem A a normal speed distribution is given in the compressor-inlet plane, in problem B we are given a pressure distribution. We look for the induced pressure and velocity fields. Using the acoustical approximation of the fundamental equations the Wiener-Hopf-method is applied then leading, after a multiplicative and additive decomposition procedure, to an infinite system of linear equations. These involve the expansion coefficients of the solution with respect to the eigenfunctions of the concentric cylindrical duct. Applying arguements from perturbation theory it is possible to extract the essential information from this system of equations. Apart from the possible exception of a denumerable set of values of the distance L of the blade-row from the inlet the problem has a unique solution.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 9
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: As, in general, the projections of characteristics into the x-space intersect for finite values of t, the global solution of a conservation law cannot be determined from the characteristic system of the equation, is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convex h and all x0 this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristics.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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