Laplace transformation

The history of electrochemical impedance spectroscopy (EIS) is briefly reviewed, starting with the foundations laid by Heaviside in the late 19th century in the form of Linear Systems Theory (LST). Warburg apparently was the first to extend the concept of impedance to electrochemical systems at the turn of the 19th century, when he derived the impedance function for a diffusional process that still bears his name. Impedance spectroscopy was next employed extensively using reactive bridges to measure the capacitance of ideally polarizable electrodes (mostly mercury), leading to the development of models for the electrified interface. However, it was the invention of the potentiostat in the 1940s and the development of frequency response analyzers in the 1970s that led to the use of EIS in exploring electrochemical and corrosion mechanisms, primarily because of their ability to probe electrochemical systems at very low frequencies. These inventions have led to an explosion in the use of EIS for exploring a wide range of systems and processes, ranging from conduction in the solid and liquid states, ionic and electronic conduction in polymers, heterogeneous reaction mechanisms, and the important phenomenon of passivity. It is evident that the use of EIS in identifying reaction mechanisms makes use of pattern recognition, currently through inspection. It is argued that, in the future development of EIS, reaction mechanism analysis (RMA) would be most efficiently done by using artificial neural networks operating in the pattern recognition mode. This strategy would require the creation of libraries of reaction mechanisms for which the theoretical impedance functions are known.

A semi-analytical analysis for the transient elastodynamic response of an arbitrarily thick simply supported beam due to the action of an arbitrary moving transverse load is presented, based on the linear theory of elasticity. The solution of the problem is derived by means of the powerful state space technique in conjunction with the Laplace transformation with respect to the time coordinate. The inversion of Laplace transform has been carried out numerically using Durbin’s approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations and obtain uniformly convergent solutions. Detailed numerical results for the transient vibratory responses of concrete beams of selected thickness parameters are obtained and compared for three types of harmonic moving concentrated loads: accelerated, decelerated and uniform. The effects of the load velocity, pulsation frequency and beam aspect ratio on the dynamic response are examined. Also, comparisons are made against solutions based on Euler–Bernoulli and Timoshenko beam models. Limiting cases are considered, and the validity of the model is established by comparison with the solutions available in the existing literature as well as with the aid of a commercial finite element package.

A technique is described for the solution of the wave equation with time dependent boundary conditions. The finite element solution accompanied by the numerical Laplace inversion process Seems to be an efficient procedure to treat such problems. 'The programming involved is straightforward in the sense that numerical Laplace inversion routines can be directly used as a time integration procedure after obtaining standard finite element differential equation solutions inthe transformed domain.

In this note we present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscousdiffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain. Comparing the fractional results for boundary shear-stress and fluid speed to the existing analytical results for the first and second Stokes problems, the fractional methodology is validated and shown to be much simpler and more powerful than existing techniques. Fig. 1 Time evolution of surface shear stress, Eq. "21…, and the imposed boundary condition "dashed line…: U"t…ÕUÄsin"t….

We consider the problem of finding a function defined on (0, ∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.

Hermitian matrices and let be the cone of the positive definite ones. We say that the random variable S, taking its values in , has the complex Wishart distribution γ p,σ if E(exp trace(θS)) = (det(I r − σ θ)) −p , where σ and σ −1 − θ are in , and where p = 1, 2,. .. , r − 1 or p > r − 1. In this paper, we compute all moments of S and S −1. The techniques involve in particular the use of the irreducible characters of the symmetric group.

With increased operating frequencies and circuit component density, signal and power integrity problems caused by voltage bounces have become more important for high-speed digital systems. This paper presents a systematic macromodeling of power-bus structures based on the cavity model: first, more accurate models are used to describe conductor and dielectric losses, thus improving accuracy at low frequencies; second, a rational model of the overall power/ground structure in the Laplace domain is presented, and third, the rational macromodel is used to identify the dominant poles, and a simplified Spice-compatible equivalent circuit is synthesized by using only the selected poles. The numerical results confirm the correctness and effectiveness of the method.

Lecture Notes Contents:
Chapter 1 First-Order Differential Equations
1.2 Basic Ideas and Terminology
1.4 Separable differential equations
1.6 First-order linear differential equations
1.9 Exact Differential equations
Chapter 6  Linear Differential Equations of Order n
6.1 General theory for linear differential equations
6.9 Reduction of order
6.2 Constant-coefficient homogeneous linear differential equations
6.3 The method of undetermined coefficients
6.7 The Variation of Parameters Method
6.8 A differential equation with nonconstant coefficients
Chapter 8  The Laplace transform method
The inverse transform Laplace transform solution of linear differential equations with constant coefficients
and tutorials with different examples.

Non-linear transient heat conduction in a hollow cylinder with temperaturedependent thermal conductivity is investigated numerically by using the hybrid application of the Laplace transform technique and the finite-element method (FEM) or the finite-difference method (FDM). The governing equation is discretized in the space domain using the FEM or the FDM. and then the non-linear terms are linearized by using the Taylor series expansion. The time-dependent terms in the linearized equations are removed by the Laplace transform technique. The numerical inversion of the Laplace transform is used to invert the transformed temperature to the result in the physical quantity. Since the present method is not a time-stepping procedure, the results at a specific time can be calculated in the time domain without step-by-step computations. The computational results at a specific time are convergent over the entire space domain at about the fourth or fifth iteration. To show the accuracy of the present method, a comparison of the present results and those using the FDM in conjunction with the Crank-Nicolson algorithm is made.

The primary objective of this project is to extend the conveniences of
deconvolution to non-linear problems of fluid flow in porous media. Unlike
conventional approaches, which are based on an approximate linearization of the
problem, here the solution of the non-linear problem is linearized by a perturbation
approach, which permits term-by-term application of deconvolution. Because the
proposed perturbation solution is more conveniently evaluated in the Laplacetransform
domain and the standard deconvolution algorithms are in the time-domain,
an efficient deconvolution procedure in the Laplace domain is a prerequisite.
For this research objective, a new algorithm is introduced which uses inverse
mirroring at the points of discontinuity and adaptive cubic splines to approximate rate
or pressure versus time data. This algorithm accurately transforms sampled data into
Laplace space and eliminates the Numerical inversion instabilities at discontinuities
or boundary points commonly encountered with the piece-wise linear approximations
of the data.
Applying the algorithm to the field data obtained from published works, we can
unveil the early-time behavior of a reservoir system masked by wellbore-storage
effects. The wellbore-storage coefficient can be variable in the general case. The new
method thus provides a powerful tool to improve pressure-transient-test
interpretation.
Practical use of the algorithm presented in this research has applications in a
variety of Pressure Transient Analysis (PTA) and Rate Transient Analysis (RTA)
problems. A renewed interest in this procedure is inspired from the need to evaluate
production performances of wells in unconventional reservoirs. With this approach,

The model of the two-dimensional equations of generalized thermo-viscoelasticity with two relaxation times is established. The state space formulation for two-dimensional problems is introduced. Laplace and Fourier integral transforms are used. The resulting formulation is applied to a problem of a thick plate subject to heating on parts of the upper and lower surfaces of the plate that varies exponentially with time. The Fourier transforms are inverted analytically. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for the problem considered. Comparisons are made with the results predicted by the coupled theory. Ó

In this paper, the Laplace Decomposition Method (LDM) employed to obtain approximate analytical solutions of the Klein-Gordon equation. The results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme.

The research reported herein involves the study of the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include cases where: (a) one (or both) spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque.

In this work, we present a review of the GILTT (Generalized Integral Laplace Transform Technique) solutions for the one and two-dimensional, time-dependent, advection-diffusion equations focusing the application to pollutant dispersion simulation in atmosphere, assuming both Fickian and counter-gradient models for a wide class of problems. For sake of completeness, we also report numerical simulations and statistical comparisons with experimental data and results of literature.

This work is aiming to present an analytical method to study the dynamic behavior of thermoelastic stresses in a finite-length functionally graded (FG) thick hollow cylinder under thermal shock loading. The thermo-mechanical properties are assumed to vary continuously through the radial direction as a nonlinear power function. Using Laplace transform and series solution, the thermoelastic Navier equations in displacement form are solved analytically. The solution of the displacement field in the FG cylinder is obtained in the Laplace domain. Also, the fast Laplace inverse transform method (FLIT) is employed to transfer the results from Laplace domain to time domain. The effects of thermal shock loading on the dynamic characteristics of the FG thick hollow cylinder are studied in various points across the thickness of cylinder for various grading patterns of FGMs. A good agreement can be seen in the comparison of the obtained results based on the presented analytical method with published data.

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on ½0; 1 2 . In particular: (i) we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen-Loe`ve expansions, (iii) we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75-87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177-193]. Our results extend some classical computations due to Le´vy [Wiener's random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171-186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen-Loe`ve expansions for weighted Wiener processes and Brownian bridges via

The problems of systems identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haar and many other functions. In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann-Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.

The model of the equation of generalized thermo-viscoelasticity with two relaxation times is established. The state space formulation for thermo-viscoelasticity with two relaxation times is introduced. The formulation is valid for problems with or without heat sources. The ...

The generalized coupled thermoelasticity based on the Lord–Shulman (LS) model that admits the second sound effect is considered to study the dynamic thermoelastic response of functionally graded annular disk. The disk material is considered to be graded along the radial direction, where a power law distribution is assumed. The thermal and mechanical waves propagation are investigated applying a sudden thermal exposure to the boundary. The Laplace transform is used to transform the governing equations into the Laplace domain, where the Galerkin finite element method is employed to solve the system of equations in space domain. Finally, the numerical inversion of the Laplace transform is used to obtain the actual physical quantities in the time domain. The time dependent temperature, radial displacement, radial stress, and hoop stress are plotted.

The hyperbolic heat conduction process in a hollow sphere with its two boundary surfaces subject to sudden temperature changes is solved analytically by means of integration transformation. An algebraic analytical expression of the temperature profile is obtained. Accordingly, the non-Fourier hyperbolic heat propagation in hollow spherical medium is analyzed and possible hyperbolic anomalies are discussed.

In this article, a coupling of Laplace transformation and Differential transform method is presented for solving heat-like and wave-like equations with variable coefficients. We demonstrate that the proposed method is very convenient for achieving the analytical solutions of 2D and 3D partial differential equations. The numerical computation shows the efficiency and simplicity of the method.

A detailed study is undertaken to analyze the non-steady interaction of plane progressive pressure pulses with an isotropic, homogeneous, fluid-filled and submerged spherical elastic shell of arbitrary wall thickness within the scope of linear acoustics. The formulation is based on the general three dimensional equations of linear elasticity and the wave equation for the internal and external acoustic domains. The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables is used to obtain the transformed solutions in the form of partial-wave expansions in terms of Legendre polynomials. The inversion of Laplace transforms have been carried out numerically using Durbin's approach based on Fourier series expansion. Special convergence enhancement techniques are invoked to completely eradicate spurious oscillations (Gibbs' phenomenon), and obtain uniformly convergent solutions. Detailed numerical results for the transient and vibratory responses of water-submerged steel shells of selected wall thickness parameters with various internal fluid loadings under an exponential wave excitation are presented. Many of the interesting dynamic features in the transient shell-shock interaction such as shock transparency, shell-radiated negative pressure waves, formation of triple points, and focusing of the reflected waves are examined using appropriate 2D images of the internal pressure field. Also, the temporal behavior of the specularly-reflected, the lowest symmetric S 0 -and antisymmetric A 0 -Lamb waves, as well as appearance of the Franz's creeping waves are discussed through proper visualization of the external scattered field. Likelihood of cavitation is addressed and regions proned to cavitation are identified. Moreover, the effects of internal fluid impedance in addition to shell wall thickness on the dynamic stress concentrations induced within the shell are analyzed. Limiting cases are considered and fair agreements with well-known solutions are established.

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate is investigated. Exact solutions for the velocity field are established by means of the Fourier and Laplace transforms. The similar solutions for Maxwell and Newtonian fluids can be obtained as limiting cases of our results. In the absence of side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.

In this paper we present a Laplace transform-based analytical solution for pricing double-barrier options under a flexible hyper-exponential jump diffusion model (HEM). The major theoretical contribution is that we prove non-singularity of a related high-dimensional matrix, which guarantees the existence and uniqueness of the solution.

In this paper we explore discrete monitored barrier options in the Black-Scholes framework. The discontinuity arising at each monitoring data requires a careful numerical method to avoid spurious oscillations when low volatility is assumed. Here a technique mixing the Laplace Transform and the finite difference method to solve Black-Scholes PDE is considered. Equivalence between the Post-Widder inversion formula joint with finite difference and the standard finite difference technique is proved. The mixed method is positivity-preserving, satisfies the discrete maximum principle according to financial meaning of the involved PDE and converges to the underlying solution. In presence of low volatility, equivalence between methods allows some physical interpretations.

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω of positive definite real matrices. The highest weight representations of the group Sp(n, R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω + iSym(n, R). We then use the Laplace transform to carry the Lie algebra action over to L 2 (Ω, dµν ). The differential recursion relations result by restricting to a distinguished three dimensional subalgebra, which is isomorphic to sl(2, R).

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace-Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.

We revisit the Cauchy problem for the time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order b 2 ð0; 2. By using the Fourier-Laplace transforms the fundamentals solutions (Green functions) are shown to be high transcendental functions of the Wright-type that can be interpreted as spatial probability density functions evolving in time with similarity properties. We provide a general representation of these functions in terms of Mellin-Barnes integrals useful for numerical computation.

The tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where, even in the time-varying case, a (formal) Laplace transform and the transfer matrix are most naturally defined. Several classic problems are examined in this algebraic setting: the relationship between left (right) coprime matrix decomposition and controllability (observability), the state-variable canonical realization, the transfer algebra with respect to parallel and series connections, the input-output inversion, and model matching.

Using two kinds of multivariate regular variation we prove several Abel-Tauber theorems for the Laplace transform of functions in several variables. ,We generalize some power series results of Alpar and apply our results in multivariate renewal theory.

Resiniferatoxin (RTX) is an ultrapotent capsaicin analog that binds to the transient receptor potential channel, vanilloid subfamily member 1 (TRPV1). There is a large body of evidence supporting a role for TRPV1 in noxious-mediated and inflammatory hyperalgesic responses. In this study, we evaluated low, graded, doses of perineural RTX as a method for regional pain control. We hypothesized that this approach can provide long-term, but reversible, blockade of a portion of nociceptive afferent fibers within peripheral nerves when given at a site remote from the neuronal perikarya in the dorsal root ganglia. Following perineural RTX application to the sciatic nerve, we demonstrated a significant inhibition of inflammatory nociception that was dose-and time-dependent. At the same time, treated animals maintained normal proprioceptive sensations and motor control, and other nociceptive responses were largely unaffected. Using a range of mechanical and thermal algesic tests, we found that the most sensitive measure following perineural RTX administration was inhibition of inflammatory hyperalgesia. Recovery studies showed that physiologic sensory function could return as early as two weeks post-RTX treatment, however, immunohistochemical examination of the DRG revealed a partial, but significant reduction in the number of the TRPV1-positive neurons. We propose that this method could represent a beneficial treatment for a range of chronic pain problems, including neuropathic and inflammatory pain not responding to other therapies.

The micromechanical method of cells is used to calculate the average time-dependent constitutive properties of the homogenized substitute continuum from the viscoelastic material properties and volume fractions of the individual phases as well as from the interphase behavior of polymeric fiber composites. Two approaches are investigated to establish and solve the micromechanical model equations of the rate-dependent material phases. The first one is based on the LAPLACE APLACE-transformation of the time-dependent material functions and the application of the correspondence principle of linear viscoelasticity to the governing equations of the micromechanical model. In the second approach a numerical time integration scheme is developed to compute the convolution integrals. For the study of the influence of compliant fiber-matrix bonds alternative models are proposed: A representative volume element with three constituents accounts for the properties of the flexible interphase between the fibers and the matrix. The three-phase model is then compared to a two-phase model with a time-dependent flexible interface to simulate the compliant fiber-matrix bond. Various examples show the performance of both models and the numerical algorithms presented.

In this paper the Parseval theorem for Laplace and Stieltjes transforms which was proved by Yurekli (1989, IMA J. Appl. Math., 42, 241-249) for conventional functions is proved for generalized functions. The theorem yields some corollaries similar to the corollaries in Yurekli (1989).

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The probability distribution of the cascade generators in a random multiplicative cascade represents a hidden parameter which is reflected in the fine scale limiting behavior of the scaling exponents (sample moments) of a single sample cascade realization as a.s. constants. We identify a large class of cascade generators uniquely determined by these scaling exponents. For this class we provide both asymptotic consistency and confidence intervals for two different estimators of the cumulant generating function (log Laplace transform) of the cascade generator distribution. These results are derived from investigation of the convergence properties of the fine scale sample moments of a single cascade realization.

We present a new procedure for the numerical calculation of the transient response of systems characterized by partial differential equations in several space variables and time. The procedure is based on: (i) spatially discretizing the system, (ii) deriving an equivalent circuit for each volume element, (iii) applying a time Laplace transformation and obtaining an admittance matrix, (iv) writing the expressions for the Laplace transforms of the desired variables according to Cramer's determinant rule, (v) interpreting the determinants as generalized eigenvalue problems, (vi) obtaining the proper eigenvalues of the system (poles and zeros), and (vii) finding the time-dependent solutions by 'trivial' inverse Laplace transformations. Although the procedure proposed applies to many partial differential equations (fluid flow in porous media, conductive heat transfer with convection, pressure tests in oil wells, electrical conduction, etc.), in the paper we illustrate the procedure in detail for a conduction heat transfer system (nine nodes). The procedure presented is much more efficient numerically than the direct numerical integration (in space and time) of the ruling PDE, or than solutions which require 'standard' numerical inverse Laplace transformation after matrix inversion.

In this paper, the Laplace Decomposition Method (LDM) employed to obtain approximate analytical solutions of the Klein-Gordon equation. The results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme.