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# Integral Transformations, Operational Calculus, and Generalized Functions

31 Jul 1996-

TL;DR: The Titchmarsh Theorem as discussed by the authors describes the order of infinite processes in Laplace Transformations and Inverse Two-Dimensional Laplacians, which is a special case of Fourier Transformations.

Abstract: Preface. 1. Laplace Transformations. 2. Mikusinski Operators. 3. Fourier Transformations. 4. Generalized Functions. 5. Other Transformations. References. Appendices: A. The Titchmarsh Theorem. B. Inversion Integrals. C. Interchange of Order of Infinite Processes. D. Definitions and Properties of Some Special Functions. Tables of Transforms: 1. Laplace. 2. Inverse Laplace. 3. Fourier. 4. Fourier Cosine. 5. Fourier Sine. 6. Mellin. 7. Power Series. 8. Finite Fourier. 9. Finite Fourier Cosine. 10. Finite Fourier Sine. 11. Finite Laplace. 12. Two-Dimensional Laplace. 13. Inverse Two-Dimensional Laplace. Index.

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TL;DR: In this article, the steady-state response of a uniform infinite Euler-Bernoulli elastic beam resting on a Pasternak elastic foundation and subjected to a concentrated load moving at a constant velocity along the beam is analytically investigated.

46 citations

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TL;DR: In this article, a finite element method approach coupled with a direct integration algorithm is developed for efficiently tracing the nonlinear dynamic response of the beam-foundation system under the action of a transverse concentrated load, moving at a constant velocity along the beam, displaying an harmonic-varying magnitude in time.

Abstract: The present paper is concerned with the numerical modelization of the transient dynamic response of a simply supported Euler–Bernoulli elastic beam resting on a Winkler-type foundation, under the action of a transverse concentrated load, moving at a constant velocity along the beam, displaying an harmonic-varying magnitude in time. The elastic foundation, assumed as homogeneous in space, behaves according to a bilinear constitutive law, characterized by two different stiffness coefficients in compression and in tension. A finite element method approach coupled with a direct integration algorithm is developed for efficiently tracing the nonlinear dynamic response of the beam-foundation system. An original automated procedure is set, as being apt to resolve all required space/time discretization issues. Extensive parametric numerical analyses are performed to investigate how the frequency of the harmonic moving load amplitude and the ratio between the foundation’s moduli in compression and in tension affect the so-called critical velocities of the moving load, leading to high transverse beam deflections. Analytical interpolating expressions are proposed and fitted for the achieved two-branch critical velocity trends. The present outcomes shall reveal potential practical implications in scenarios of contemporary railway engineering, especially in terms of lowering down the admissible high-speed train velocities, as for structural requirement or preventing potential passenger discomfort.

28 citations

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21 Mar 2018

TL;DR: In this paper, structural dynamics analysis of one-dimensional elements (strings, beams) on continuous elastic support under high-velocity moving load is the main subject of the present doctoral dissertation.

Abstract: The structural dynamics analysis of one-dimensional elements (strings, beams) on continuous elastic support under high-velocity moving load is the main subject of the present doctoral dissertation. Two main types of mechanical systems have been considered, of a finite and of an infinite extension. Through ad hoc formulations and autonomous implementations, physical dynamic response characteristics of taut string/beam-foundation systems are revealed by virtue of analytical and numerical approaches, both in the linear and in the nonlinear regimes.
First, two explicit closed-form analytical solutions relative to the static deflection of a finite Euler-Bernoulli elastic beam lying on aWinkler elastic foundation
with space-dependent stiffness coefficient are derived. Then, a FEM implementation is developed to investigate the transient dynamic response of a simply-supported Euler-Bernoulli beam resting on spatially homogeneous Winkler nonlinear elastic foundations under the action of a transverse concentrated
moving load, with a constant velocity and harmonic-varying magnitude in time.
Regarding the analysis of infinite systems, the steady-state responses of a uniform infinite taut string and of a uniform infinite Euler-Bernoulli elastic beam, both resting on an elastic support and subjected to a concentrated transverse moving load, are numerically obtained by an original Discontinuous Least-Squares Finite Element Method (DLSFEM) and by effective Perfectly-Matched Layer (PML) implementations. In particular, concerning the steady-state
response of the beam a wholly new, Perfectly Matched Layer (PML) for the underlying fourth-order differential problem is analytically formulated and implemented. In addition, a universal closed-form analytical solution is derived for the infinite beam moving load problem, apt to represent the response for all possible beam-foundation parameters.
The present thesis demonstrates the reliability and effectiveness of all the derived analytical-numerical solutions, through extensive parametric analyses, carried out for interpreting the parametric variation of the mechanical response of the considered systems due to changes in their characteristic mechanical properties.

9 citations

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08 Sep 2011

TL;DR: In this paper, the authors studied the generalization of reaction-subdiffusion schemes to subdiffusion by means of continuous-time random walks on a mesoscopic scale with a heavy-tailed waiting time pdf ψ(t) ∝ t−1−α lacking the first moment.

Abstract: The present work studies the generalization of reaction–diffusion schemes to subdiffusion. The subdiffusive dynamics was modelled by means of continuous–time random walks on a mesoscopic scale with a heavy–tailed waiting time pdf ψ(t) ∝ t−1−α lacking the first moment. The reaction itself was assumed to take place on a microscopic scale, obeying the classical mass action law. This situation is assumed to apply in a porous medium where the particles are trapped within the catchments, pores and stagnant regions of the flow, but are still able to react during their waiting times. After discussing the subdiffusion equation and different methods of their solution, especially under the aspect of particles being introduced into the system in the course of time, the reaction–subdiffusion equations are addressed. These equations are of integro–differential form and under the assumptions made, the reaction explicitly affects the transport term. The long ranged memory of the subdiffusion kernel is modified by an additional factor accounting for the conversion and survival probabilities due to reaction during the waiting times. In the case of linear reaction kinetics, this factor is governed by the rate coefficients. For nonlinear reaction kinetics the transport kernel depends additionally on the concentrations of the respective reaction partners at all previous times. The simplest linear reaction, the degradation A→ 0 was considered and a general expression for the solution to arbitrary Dirichlet Boundary Value Problems was derived. This solution can be expressed in terms of the solution to the corresponding Dirichlet Problem under mere subdiffusion, i.e. without degradation. The resultant stationary profiles do not differ qualitatively from the stationary profiles in normal reaction diffusion. For stationary solutions to exist in reaction–subdiffusion, the assumption of reactions according to classical rate kinetics is essential. As an example for a nonlinear reaction–subdiffusion system, the irreversible autocatalytic reaction A + B→ 2A under subdiffusion is considered. Under the assumptions of constant overall particle concentration A(x, t) + B(x, t) = const and re–labelling of the converted particles, a subdiffusive analogue of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation was derived and the resultant fronts of A–particles propagating into the B– domain were studied. Two different regimes were detected in numerical simulations. These regimes were discussed using both crossover arguments and analytic calculations. The first regime can be described within the framework of the continuous reaction–subdiffusion equations and is characterized by the front velocity and width going as t α−1 2 at larger times. As the front width decays, the front gets atomically sharp at very large times and a transition to a second regime, the fluctuation dominated one, is expected. The fluctuation dominated regime is not within the scope of the continuous description. In that case, the velocity of the front decays faster in time than in the continuous regime, v f luct ∝ tα−1. Further simulations pertaining the reaction on contact scenario, i.e. the fluctuation dominated regime, revealed additional fluctuation effects that are genuinely due to subdiffusion. Another nonlinear reaction–subdiffusion system where reactants A penetrate a medium initially filled with immobile reactants B and react according to the scheme A+B→ (inert) was considered. Under certain presumptions, this problem can be described in terms of a moving boundary problem, a so–called Stefan–problem, for the concentration of a single species. The main result was that the propagation of the moving boundary between the A– and B–domain goes as R(t) ∝ tα/2. The theoretical predictions concerning the moving boundary were corroborated by numerical simulations.

6 citations