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  • 1
    Electronic Resource
    Electronic Resource
    Some comments on the use of radial basis functions in the dual reciprocity method (1998)
    Springer
    Computational mechanics 22 (1998), S. 61-69 
    Details
    ISSN: 1432-0924
    Keywords: Key words DRM; thin plate spline; multiquadrics; the method of fundamental solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004660050339
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  • 2
    Electronic Resource
    Electronic Resource
    Particular solutions of Helmholtz-type operators using higher order polyhrmonic splines (1999)
    Springer
    Computational mechanics 23 (1999), S. 411-419 
    Details
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract We obtain explicit analytical particular solutions for Helmholtz-type operators, using higher order splines. These results generalize those in Golberg, Chen and Rashed (1998) and Chen and Rashed (1998) for thin plate splines. This enables one to substantially improve the accuracy of algorithms for solving boundary value problems for Helmholtz-type equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004660050420
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  • 3
    Electronic Resource
    Electronic Resource
    Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method (1998)
    Springer
    Computational mechanics 21 (1998), S. 141-148 
    Details
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris et al. (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004660050290
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  • 4
    Electronic Resource
    Electronic Resource
    Convergence of an initial-value method for Fredholm integral equations (1973)
    Springer
    Journal of optimization theory and applications 12 (1973), S. 344-354 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A convergence theorem is established for a method of solving Fredholm integral equations due to Kalabaet al. To do this, an interpolation procedure is developed for obtaining an approximation of the resolvent kernel. The results of Anselone and Moore on collectively compact operators are then applied to show convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940416
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  • 5
    Electronic Resource
    Electronic Resource
    An initial-value method for the determination of Green's functions (1973)
    Springer
    Journal of optimization theory and applications 11 (1973), S. 506-516 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935663
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  • 6
    Electronic Resource
    Electronic Resource
    Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels (1978)
    Springer
    Journal of optimization theory and applications 24 (1978), S. 89-131 
    Details
    ISSN: 1573-2878
    Keywords: Integral equations ; Fredholm equations ; boundary-value problem ; numerical solution ; Cauchy problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933183
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  • 7
    Electronic Resource
    Electronic Resource
    Numerical solution of a class of integral equations arising in two-dimensional aerodynamics (1978)
    Springer
    Journal of optimization theory and applications 24 (1978), S. 169-206 
    Details
    ISSN: 1573-2878
    Keywords: Aerodynamics ; integral equation of the first kind ; singular kernels ; Galerkin's method ; collocation method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This enables us to study the problem in terms of an equivalent integral equation of the second kind. Using Galerkin's method, we are able to derive a convergent numerical algorithm for its solution. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as our method of computation. Extensive numerical calculations are presented establishing the validity of the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933186
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  • 8
    Electronic Resource
    Electronic Resource
    On a method of Bownds for solving Volterra integral equations (1978)
    Springer
    Journal of optimization theory and applications 24 (1978), S. 221-232 
    Details
    ISSN: 1573-2878
    Keywords: Integral equations ; numerical methods ; differential equations ; numerical integration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An initial-value method of Bownds for solving Volterra integral equations is reexamined using a variable-step integrator to solve the differential equations. It is shown that such equations may be easily solved to an accuracy ofO(10−8), the error depending essentially on that incurred in truncating expansions of the kernel to a degenerate one.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933188
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  • 9
    Electronic Resource
    Electronic Resource
    A generalized Rayleigh quotient for eigenvalue problems nonlinear in the parameter (1973)
    Springer
    Journal of optimization theory and applications 11 (1973), S. 146-158 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A functional is given which generalizes the Rayleigh quotient to eigenvalue problems for linear operators where the eigenvalue parameter appears nonlinearly. Particular emphasis is given to the development of perturbation-type results for eigenvalues and characteristic values which generalize the classical results. Applications are made to eigenvalue and characteristic value problems for integral and matrix operators and to the critical length problem for integral operators. Both symmetric and nonsymmetric operators are treated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935880
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  • 10
    Electronic Resource
    Electronic Resource
    The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations (1998)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 43 (1998), S. 1421-1435 
    Details
    ISSN: 0029-5981
    Keywords: method of fundamental solutions ; particular solution ; diffusion equations ; quasi-Monte-Carlo method ; Laplace transform ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The Laplace transform is applied to remove the time-dependent variable in the diffusion equation. For non-harmonic initial conditions this gives rise to a non-homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we find through a method suggested by Atkinson. To avoid costly Gaussian quadratures, we approximate the particular solution using quasi-Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand. The approximate transformed solution is then inverted numerically using Stehfest's algorithm. Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2-D and 3-D. © 1998 John Wiley & Sons, Ltd.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
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