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Sinc methods for computing solutions to viscoelastic and related problems. (English) Zbl 1334.65211

Summary: Sinc methods enable approximations to every operation of calculus, including the solution of differential and integral equations, and in this paper we illustrate some of these techniques to solve viscoelastic problems of the type considered by most engineers.
For example, in one dimension, sinc methods enable the following, over finite, semi-infinite, infinite intervals or arcs: interpolation, differentiation, definite and indefinite integration, definite and indefinite convolution, Hilbert and Cauchy transforms, evaluation and inversion of Laplace transforms, solution of ordinary differential equation initial value problems, and solution of convolution-type integral equations.
In more than one dimension, sinc methods enable accurate solutions to nonlinear elliptic, hyperbolic, and parabolic partial differential equations (PDE), as well as integral equations and conformal map problems. The regions for these problems can be curvilinear, finite, or infinite. The rate of convergence of the programs of SINC-PACK is usually orders of magnitude faster than that of current (e.g., finite element, or finite difference) methods.{ }Most numerical methods for solving partial differential equations (PDE) are based on approximation of the highest derivatives of a PDE. This is also the case for methods used in (finite difference, finite element, spectral, etc.) that are based on approximation of the highest derivatives of a PDE and thus require the solution of large systems of equations. The methods for solving PDE are based on using sinc convolution; they enable separation of variables, which require one-dimensional (i.e., small) matrices to solve PDE, even for curvilinear regions, as well as for many nonlinear equations. This occurs, in essence, when calculus is used to model the PDE. In other words, Sinc-Pack always enables solution of PDE va use of separation of variables, unlike the 1953 prediction in [P. M. Morse and H. Feshbach, Methods of theoretical physics. Vol. I. New York: McGraw-Hill Book (1953; Zbl 0051.40603)], that for solving either Poisson or Helmholtz problems in 3 dimensions, separation of variables is possible for at most 13 coordinate systems.
We illustrate here the numerical solution of some problems that are typical of the type considered in the areas of polymers and viscoelastic solids.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs

Citations:

Zbl 0051.40603

Software:

Sinc-Pack
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