Book•
Tables of Laplace Transforms
01 Jan 1973-
TL;DR: Inverse Laplace Transforms and Inverse Trigonometric Functions as mentioned in this paper have also been used to generate generalized hypergeometric functions, such as the generalized generalized Hypergeometric function (GHEF).
Abstract: I. Laplace Transforms.- 1.1 General Formulas.- 1.2 Algebraic Functions.- 1.3 Powers of Arbitrary Order.- 1.4 Sectionally Rational- and Rows of Delta Functions.- 1.5 Exponential Functions.- 1.6 Logarithmic Functions.- 1.7 Trigonometric Functions.- 1.8 Inverse Trigonometric Functions.- 1.9 Hyperbolic Functions.- 1.10 Inverse Hyperbolic Functions.- 1.11 Orthogonal Polynomials.- 1.12 Legendre Functions.- 1.13Bessel Functions of Order Zero and Unity.- 1.14 Bessel Functions.- 1.15 Modified Bessel Functions.- 1.16 Functions Related to Bessel Functions and Kelvin Functions.- 1.17 Whittaker Functions and Special Cases.- 1.18 Elliptic Functions.- 1.19 Gauss' Hypergeometric Function.- 1.20 Miscellaneous Functions.- 1.21 Generalized Hypergeometric Functions.- II. Inverse Laplace Transforms.- 2.1 General Formulas.- 2.2 Rational Functions.- 2.3 Irrational Algebraic Functions.- 2.4 Powers of Arbitrary Order.- 2.5 Exponential Functions.- 2.6 Logarithmic Functions.- 2.7 Trigonometric- and Inverse Functions.- 2.8 Hyperbolic- and Inverse Functions.- 2.9 Orthogonal Polynomials.- 2.10 Gamma Function and Related Functions.- 2.11 Legendre Functions.- 2.12 Bessel Functions.- 2.13 Modified Bessel Functions.- 2.14 Functions Related to Bessel Functions and Kelvin Functions.- 2.15 Special Cases of Whittaker Functions.- 2.16 Parabolic Cylinder Functions and Whittaker Functions.- 2.17 Elliptic Integrals and Elliptic Functions.- 2.18 Gauss' Hypergeometric Functions.- 2.19 Generalized Hypergeometric Functions.- 2.20 Miscellaneous Functions.
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TL;DR: In this article, the authors present a review of continuous-time random walk theory for diffusion of single particles on lattices with frozen-in disorder, including models with regular transition rates and irregular lattices.
835 citations
TL;DR: In this paper, an exact analytical solution for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix is developed for the same problem, taking into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes.
Abstract: An exact analytical solution is developed for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix. The solution takes into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes, molecular diffusion from the fracture to the porous matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. The general transient solution is in the form of a double integral that is evaluated using Gauss-Legendre quadrature. A transient solution is also presented for the simpler problem that assumes negligible longitudinal dispersion along the fracture. This assumption is usually reasonable when the advective flux in a fracture is large. A comparison between two steady state solutions, one with dispersion and one without, permits a criterion to be developed that is useful for assessing the significance of longitudinal dispersion in terms of the overall system response. Examples of the solutions demonstrate that penetration distances along fractures can be substantially larger through multiple, closely spaced fractures than through a single fracture because of the limited capability of the finite matrix to store solute.
626 citations
TL;DR: In this paper, it is shown how the equilibrium pair correlation function between spinbearing molecules in liquids may be incorporated as an effective force in the relative diffusion expressions, and how one may solve for the resulting time correlation functions and spectral densities needed for studies of spin relaxation by translational diffusion.
Abstract: It is shown how the equilibrium pair correlation function between spin‐bearing molecules in liquids may be incorporated as an effective force in the relative diffusion expressions, and how one may solve for the resulting time correlation functions and spectral densities needed for studies of spin relaxation by translational diffusion. The use of finite difference methods permits the solution no matter how complex the form of the pair correlation function (pcf) utilized. In particular, a Percus–Yevick pcf as well as one corrected from computer dynamics, both for hard spheres, are utilized. Good agreement with the experiments of Harmon and Muller on dipolar relaxation in liquid ethane is obtained from this analysis. Effects of ionic interactions in electrolyte solutions upon dipolar relaxation are also obtained in terms of Debye–Huckel theory for the pcf. Analytic solutions are given which are appropriate for the proper boundary‐value problem for the relative diffusion of molecules (i.e., a distance of mini...
516 citations
TL;DR: The theory of exact boundary conditions for constant coefficient time-dependent problems is developed in detail, with many examples from physical applications as discussed by the authors, and an illustrative numerical example is given.
Abstract: We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient time-dependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to compare different accurate treatments, and an illustrative numerical example is given. We close with a discussion of some important problems that remain open.
413 citations
Book•
14 Oct 1999
TL;DR: The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial as mentioned in this paper, and it can also be used to solve difference equations, such as the Riemann-Stieltjes integral.
Abstract: The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial. It can also be used to solve difference equations. The present text, while mathematically rigorous, is readily accessible to students of either mathematics or engineering. Even the Dirac delta function, which is normally covered in a heuristic fashion, is given a completely justifiable treatment in the context of the Riemann-Stieltjes integral, yet at a level an undergraduate student can appreciate. When it comes to the deepest part of the theory, the Complex Inversion Formula, a knowledge of poles, residues, and contour integration of meromorphic functions is required. To this end, an entire chapter is devoted to the fundamentals of complex analysis. In addition to all the theoretical considerations, there are numerous worked examples drawn from engineering and physics.
When applying the Laplace transform, it is important to have a good understanding of the theory underlying it, rather than just a cursory knowledge of its application. This text provides that understanding.
383 citations