Numerical Laplace Inversion Research Articles

A simple and fast method for valuing American knock-out options with rebates

In this paper, we derive an analytic formula for the American knock-out options with rebate. Rather than using a probabilistic method, we use the Laplace–Carson Transform(LCT) method to induce a simple functional equation associated with the complex problem of option pricing Partial Differential equation with free boundary. The transformed value of free boundary could be solved by applying Newton’s method. Lastly, numerical Laplace inversion techniques are used to solve for the wanted free boundary value and the options value.

Active Transient Acousto-Structural Response Control of a Smart Cavity-Coupled Circular Plate System

Abstract The transient vibroacoustic response suppression of a piezo-coupled sandwich circular plate backed by a rigid-walled cylindrical acoustic enclosure is investigated. Problem formulation is based on the linear acoustic wave theory, Kirchhoff thin plate model, fluid/structure compatibility relations, Rayleigh integra formula, and active damping control (ADC) strategy. Matlab’s Genetic Algorithm (GA) is utilized to identify and optimize the feedback controller gain parameter based on a multi-objective performance index function. Durbin’s numerical Laplace inversion scheme is then used to calculate the key acousto-structural response parameters due to a transverse impulsive shock force for selected cavity depths. Numerical simulations demonstrate satisfactory performance of adopted control methodology in effective suppression of panel displacement response and radiated external sound pressure for enclosures of shallow and moderate depths. Limiting cases are considered and accuracy of the proposed model is rigorously verified.

Analytical solutions for a soil vapor extraction model that incorporates gas phase dispersion and molecular diffusion

To greatly simplify their solution, the equations describing radial advective/dispersive transport to an extraction well in a porous medium typically neglect molecular diffusion. While this simplification is appropriate to simulate transport in the saturated zone, it can result in significant errors when modeling gas phase transport in the vadose zone, as might be applied when simulating a soil vapor extraction (SVE) system to remediate vadose zone contamination. A new analytical solution for the equations describing radial gas phase transport of a sorbing contaminant to an extraction well is presented. The equations model advection, dispersion (including both mechanical dispersion and molecular diffusion), and rate-limited mass transfer of dissolved, separate phase, and sorbed contaminants into the gas phase. The model equations are analytically solved by using the Laplace transform with respect to time. The solutions are represented by confluent hypergeometric functions in the Laplace domain. The Laplace domain solutions are then evaluated using a numerical Laplace inversion algorithm. The solutions can be used to simulate the spatial distribution and the temporal evolution of contaminant concentrations during operation of a soil vapor extraction well. Results of model simulations show that the effect of gas phase molecular diffusion upon concentrations at the extraction well is relatively small, although the effect upon the distribution of concentrations in space is significant. This study provides a tool that can be useful in designing SVE remediation strategies, as well as verifying numerical models used to simulate SVE system performance.

Thermoelectric spherical shell with fractional order heat transfer

The model of fractional magneto-thermoelasticity is applied to one-dimensional problems of a thermoelectric spherical shell subjected to an arbitrary thermal loading of its external boundary in the presence of a uniform magnetic field. By means of the Laplace transform and numerical Laplace inversion, the problems are solved. The distributions of the considered temperature, stress and displacement, are represented graphically. The theories of coupled and generalized magneto-thermoelasticity follow as limited cases. Some comparisons are shown in the figures to estimate the effects of the fractional order and ramping parameters.

Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature

In this work, a novel generalized model of photothermal theory with two-temperature thermoelasticity theory based on memory-dependent derivative (MDD) theory is performed. A one-dimensional problem for an elastic semiconductor material with isotropic and homogeneous properties has been considered. The problem is solved with a new model (MDD) under the influence of a mechanical force with a photothermal excitation. The Laplace transform technique is used to remove the time-dependent terms in the governing equations. Moreover, the general solutions of some physical fields are obtained. The surface taken into consideration is free of traction and subjected to a time-dependent thermal shock. The numerical Laplace inversion is used to obtain the numerical results of the physical quantities of the problem. Finally, the obtained results are presented and discussed graphically.

Analysis of Two-Component Non-Equilibrium Model of Linear Reactive Chromatography

This article presents semi-analytical solutions and analytical temporal moments of a two-component linear reactive lumped kinetic model incorporating irreversible ( $$A\rightarrow B$$ ) and reversible ( $$A\leftrightarrows B$$ ) reactions in a fixed-bed liquid chromatographic column. Both solid and liquid phase reactions and two sets of boundary conditions are considered. The current model equations contain a coupled system of two partial differential equations (PDEs) and two ordinary differential equations (ODEs). The solution methodology successively employs the Laplace transform and linear transformation steps to uncouple the governing set of coupled differential equations. The resulting system of uncoupled ODEs is solved by applying an elementary solution technique. The numerical Laplace inversion is employed to transform back the solutions in the actual time domain. To further analyze the effects of different kinetic parameters, statistical temporal moments are derived from the Laplace-transformed solutions. The current solutions extend and generalize our recent solutions for single-solute transport models of non-reactive liquid chromatography. For verification, the analytical results are compared with the numerical solutions of a high-resolution finite volume scheme. Several case studies of practical interest are considered. Good agreements in the results validate the correctness of semi-analytical solutions and the accuracy of proposed numerical algorithm.

Analysis of linear reactive general rate model of liquid chromatography considering irreversible and reversible reactions

A linear general rate model of two-component liquid chromatography is analyzed considering heterogenous reactions of types A→B and A⇄B. The model equations incorporate axial dispersion, external and intra particle pore diffusions, interfacial mass transfer, linear sorption kinetics, and first order heterogeneous chemical reactions. The solution methodology successively employs the Laplace transform and linear transformation steps to uncouple the governing set of coupled differential equations. The resulting system of uncoupled ODEs is solved by applying an elementary solution technique. The numerical Laplace inversion is employed to transform back the solutions in the actual time domain. The current solutions extend and generalize the recent solutions of nonreactive general rate model for single-solute transport. For validation, a high resolution finite volume scheme is implemented to obtain the numerical solutions. Different case studies are considered to verify the correctness of semi-analytical solutions and the accuracy of the numerical scheme. To further study the behavior of a chromatographic reactor, numerical temporal moments of the elution profiles are presented for both reactant and product. The derived semi-analytical solutions are useful tools to study the influence of solid phase reaction rate constant, interfacial mass transfer rate, intra particle pore diffusion, and reactant adsorption affinity on the concentration profiles.

First-Passage Time Model Driven by Lévy Process for Pricing CoCos

Contingent convertible bonds (CoCos) are typical form of contingent capital that converts into equity of issuing firm or writes down if a prespecified trigger occurs. This paper proposes a general Lévy framework for pricing CoCos. The Lévy framework indicates that the difficulty in giving closed-form expression for CoCos price is the possible introduction of the Lévy process whose first-passage time problem has not been solved. According to characteristics of new Lévy measure after the measure transform, three specific Lévy models driven by drifted Brownian motion, spectrally negative Lévy process, and double exponential jump diffusion process are proposed to give the solution keeping the form of the driving process unchanged under the measure transform. These three Lévy models provide closed-form expressions for CoCos price while the latter two possess them up to Laplace transform, whose pricing results are given by combining with numerical Fourier inversion and Laplace inversion. Numerical results show that negative jumps have large influence on CoCos pricing and the Black-Scholes model would overestimate CoCos price by simply compressing jumps information into volatility while the other two models would give more accurate CoCos price by taking jump risk into consideration.

Analytic solution for American strangle options using Laplace–Carson transforms

A strangle has been important strategy for options when the trader believes there will be a large movement in the underlying asset but are uncertain of which way the movement will be. In this paper, we derive analytic formula for the price of American strangle options. American strangle options can be mathematically formulated into the free boundary problems involving two early exercise boundaries. By using Laplace–Carson Transform(LCT), we can derive the nonlinear system of equations satisfied by the transformed value of two free boundaries. We then solve this nonlinear system using Newton’s method and finally get the free boundaries and option values using numerical Laplace inversion techniques. We also derive the Greeks for the American strangle options as well as the value of perpetual American strangle options. Furthermore, we present various graphs for the free boundaries and option values according to the change of parameters.

The transient response of a functionally graded piezoelectric rod subjected to a moving heat source under fractional order theory of thermoelasticity

ABSTRACTThe transient thermo-piezoelectric response of a functionally graded piezoelectric rod subjected to a moving heat source is investigated in the context of fractional order theory of thermoelasticity proposed by Sherief. The material properties of the functionally graded piezoelectric rod are assumed to vary exponentially along the length, except for the thermal relaxation time and the specific heat, which are taken to be constant. To solve the governing equations of the problem, Laplace transform is applied, eliminating the time effect; the analytical solutions of the displacement, stress, temperature, and electric field in Laplace domain are obtained. Subsequently, the solutions of the considered variables in time domain are obtained by numerical Laplace inversion and illustrated graphically. In calculation, the effect of the fractional order parameter on the variations of the considered variables is presented.

An improved deconvolution algorithm using B-splines for well-test data analysis in petroleum engineering

Ilk et al.’s deconvolution algorithm using B-splines involves the Laplace transformation of the convolution equation with respect to production rate and wellbore pressure based on Duhamel principle. However, for common cases, the production rate function has “discontinuity” with respect to production time; it does not satisfy the precondition that the function to be transformed by Laplace transformation should be continuous. This inherent defect may directly cause enormous amount of computational time or even the failure of the numerical Laplace inversion in the deconvolution process. Based on these concerns, a fundamentally improved deconvolution algorithm using B-splines is presented here. In the convolution equation, the wellbore pressure derivative corresponding to constant unit production rate as the target of deconvolution is still represented by weighted summation of second-order B-splines; however, the computation process of the deconvolution is kept in the level of integral in the real time space instead of the Laplace space, for the reason that there will be no continuity requirement for the production rate function in the application of Duhamel principle for the deconvolution computation problem. According to the real production rate history, a technique of piecewise analytical integration is adopted for obtaining the elements of sensitivity matrix of a linear system with respect to weight coefficients; the linear system is generated by substituting the measured wellbore pressure data and corresponding variable production rate data into the convolution equation containing B-splines. The proposed direct analytical solution method of the integration for calculating the elements of the sensitivity matrix can not only guarantee the success of the deconvolution computation, but also can largely enhance the deconvolution computation speed. Moreover, in order to further improve the computation speed, a binary search method is also applied to find which production segments (with constant production rate) the measured wellbore pressure data points locate at in the deconvolution computation process. Another linear system with respect to weight coefficients for the regularization from Ilk et al.’s deconvolution algorithm is appended in order to overcome the effect of data errors. The two linear systems are combined together as an over-determined linear system, which can be solved by the least square method. Eventually, the reconstructed wellbore pressure and its derivative by B-splines corresponding to the constant unit production rate can be obtained.Numerical experimental tests demonstrate that the improved deconvolution algorithm exhibits good accuracy, computation speed and stability of data error tolerance. And the statement on how to perform the regularization when data error exists is also made in order to deconvolve the correct wellbore pressure derivative. The improved deconvolution algorithm is also applied into an actual field example. It is found that the deconvolution results by the improved deconvolution algorithm have good agreement with the ones by Von Schroeter et al.’s deconvolution algorithm and by Levitan et al.’s deconvolution algorithm as a whole; and the feature of typical log-log curves of the wellbore pressure drop and the wellbore pressure derivative corresponding to the improved algorithm is very close to the one of typical log-log curves calculated directly from the wellbore pressure data in the well shut-in period. In addition, through many numerical experimental tests, it is also concluded that as the quantity of data largely increases, the improved Ilk et al.'s deconvolution algorithm exhibits the big advantage in fast computational speed over von Schroeter et al.’s algorithm and Levitan et al.’s algorithm.

Fractional heat conduction equation for an infinitely generalized, thermoelastic, long solid cylinder

In this work, we consider the one-dimensional problem for an infinitely long solid cylinder in the context of the theory of generalized thermoelasticity with one relaxation time. The heat conduction equation with the Caputo fractional derivative of order α is used. The curved surface of the cylinder is assumed to be in contact with a rigid surface and is subjected to constant heat flux. By means of the Laplace transform and numerical Laplace inversion the problem is solved. Numerical computations for the temperature, displacement and stress distributions are carried out and displayed graphically as well as the results are discussed comprehensively.

Numerical Laplace inversion in problems of elastodynamics: Comparison of four algorithms

The objective of this work is to find a suitable algorithm for numerical Laplace inversion which could be used for effective and precise solution of elastodynamic problems. For this purpose, the capabilities of four algorithms are studied using three transforms resulted from analytical solutions of longitudinal waves in a thin rod, flexural waves in a thin beam and plane waves in a strip. In particular, the Gaver–Stehfest algorithm, the Gaver–Wynn’s rho algorithm, the Fixed-Talbot algorithm and the FFT algorithm combined with Wynn’s epsilon accelerator are tested. The codes written in Maple 16 employing multi-precision computations are presented for each method. Given the results obtained, the last mentioned algorithm proves to be the best. It is most efficient and it gives results of reasonable accuracy nearly for all tested times ranging from 3×10−7s to 3 × 103 s.

Normal impact response of a saturated porous cylinder with a penny-shaped crack

The transient response of a saturated porous cylinder containing a penny-shaped crack and subjected to a suddenly applied normal loading is investigated. The general field equations of Biot’s theory are applied in the formulation in conjunction with the Laplace and integral transforms to reduce the mixed boundary-value problem to the solution of Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used, and the time-dependent magnitudes of the local stress field near the crack border measured in terms of the dynamic stress-intensity factor are computed for selected material property values. These results are presented and compared to the results of dry medium to find out the influence of pore fluid and its extent on the dynamic stress-intensity factor.

A novel method for analytically solving a radial advection-dispersion equation

An analytical solution for solute transport in a radial flow field has a variety of practical applications in the study of the transport in push-pull/divergent/convergent flow tracer tests, aquifer remediation by pumping and aquifer storage and recovery. However, an analytical solution for radial advective-dispersive transport has been proven very difficult to develop and relatively few in subsurface hydrology have made efforts to do so, because variable coefficients in the governing partial differential equations. Most of the solutions for radial advective-dispersive transport presented in the literature have generally been solved semi-analytically with the final concentration values being obtained with the help of a numerical Laplace inversion. This study presents a novel solution strategy for analytically solving the radial advective-dispersive transport problem. A Laplace transform with respect to the time variable and a generalized integral transform technique with respect to the spatial variable are first performed to convert the transient governing partial differential equations into an algebraic equation. Subsequently, the algebraic equation is solved using simple algebraic manipulations, easily yielding the solution in the transformed domain. The solution in the original domain is ultimately obtained by successive applications of the Laplace and corresponding generalized integral transform inversions. A convergent flow tracer test is used to demonstrate the robustness of the proposed method for deriving an exact analytical solution to the radial advective-dispersive transport problem. The developed analytical solution is verified against a semi-analytical solution taken from the literature. The results show perfect agreement between our exact analytical solution and the semi-analytical solution. The solution method presented in this study can be applied to create more comprehensive analytical models for a great variety of radial advective-dispersive transport problems.

Analysis of general rate model of linear chromatography considering finite rates of the adsorption and desorption steps

This article presents semi-analytical solutions and temporal moments of the general rate model of chromatography with a focus on evaluating the effect of finite rates for the adsorption and desorption steps, typically considered to be in equilibrium. The model equations are analytically solved in the Laplace domain and numerical Laplace inversion is applied to get back solutions in the actual time domain. The expression of first four temporal moments are derived from the analytical solutions in the Laplace domain. The derived analytical solutions and moments are helpful tools to predict dynamic behaviors inside the column and to evaluate the influence of model parameters on the elution profiles, in particular the effect of finite rates of the intrinsic adsorption and desorption steps. The correctness of analytical solutions are verified through the numerical solutions of a high resolution finite volume scheme. Several case studies are considered to quantify effects of the rate constants for adsorption and desorption, axial dispersion, film mass transfer resistance, intraparticle diffusion resistance, and inlet boundary conditions on the elution profiles.

Analysis of a Two-Dimensional Nonequilibrium Model of Linear Reactive Chromatography Considering Irreversible and Reversible Reactions

This article presents semianalytical solutions and analytical temporal moments of a two-dimensional nonequilibrium transport model of linear reactive chromatography considering irreversible (A → B) and reversible (A ⇌ B) reactions. The model is formed by a system of four coupled partial differential equations accounting for linear advection, longitudinal and radial dispersions, the rate of variation of the local concentration of each component in the stationary phase, local deviation from equilibrium concentrations, and first-order chemical reactions in both liquid and solid phases. The solution process successively employs Hankel, Laplace, and linear transformation steps to uncouple the governing set of coupled differential equations. The resulting uncoupled systems of ordinary differential equations are solved using an elementary solution technique. The numerical Laplace inversion is applied for back-transformation of the solutions in the actual time domain. To analyze the effects of different kinetic p...

The robust identification of exchange from T2–T2 time-domain features

Two-dimensional spin–spin relaxation (T2–T2) techniques have been developed to probe coupling between different environments such as diffusive coupling between small and large pores or chemical exchange with clays. In these studies, Numerical Laplace Inversion (NLI) is used to obtain two-dimensional T2–T2 relaxation distribution spectrum from the T2–T2 signal decays, and the off-diagonal peaks characterize coupling. Often, these coupling peaks are both weak and close to the diagonal and combined with the inherently ill-conditioned nature of the inversion, their presence is difficult to differentiate from inversion related artifacts and blurring. This manuscript presents a time domain based analysis to identify the presence of coupling that avoids the ambiguities present in T2–T2 spectra. The approach utilizes the symmetric nature of the two-dimensional time domain data, where the presence of curvature along one of these symmetries gives an unambiguous indicator of coupling. Measurements on porous glass beads are used to verify the technique.

Pricing turbo warrants under mixed-exponential jump diffusion model

Turbo warrant is a special type of barrier options in which the rebate is calculated as another exotic option. In this paper, using Laplace transforms we obtain the valuation of turbo warrant under the mixed-exponential jump diffusion model, which is able to approximate any jump size distribution. The numerical Laplace inversion examples verify that the analytical solutions are accurate. The results of simulation confirm the argument that jump risk should not be ignored in the valuation of turbo warrants.

Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy

In this paper, we use a Markov-modulated regime switching approach to model various states of the economy, and study the pricing of vulnerable European options when the dynamics of the underlying asset value and the asset value of the counterparty follow two correlated jump-diffusion processes under regime switching. The correlation is modelled by both the diffusion parts and the pure jump parts which describe the uncertainty of the value of the risky assets. We develop a method to determine an equivalent martingale measure and a parsimonious representation of the risk-neutral density is provided. Based on this, we derive an analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, and implement the formula through numerical Laplace inversion.