The Numerical Inversion of the Laplace Transform in a Multi-Precision Environment

This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

THE TRANSIENT TWO-DIMENSIONAL FLOW THROUGH DOUBLE POROUS MEDIA

This paper presents the analytical solutions in Laplace domain for two-dimensional nonsteady flow of slightly compressible liquid in porous media with double porosity by using the methods of integral transforms and variables separation. The effects of the ratio of storativities to , interporosity flow parameter on the pressure behaviors for a vertically fractured well with infinite conductivity are investigated by using the method of numerical inversion. The new log-log diagnosis graph of the pressures is given and analysed.

A DISCRETE ALGORITHM FOR COMPLEX FREQUENCY-DOMAIN CONVOLUTIONS

A discrete algorithm suitable for the computation of complex frequency-domain convolution on computers was derived. The Durbin's numerical inversion of Laplace transforms can be used to figure out the time-domain digital solution of the result of complex frequency-domain convolutions. Compared with the digital solutions and corresponding analytical solutions, it is shown that the digital solutions have high precision.

Numerical Inversion for the Initial Distribution in the Multi-TermTime-FractionalDiffusion Equation Using Final Observations

This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations.The inversion problem is of instability,but it is uniquely solvable based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function.From view point of optimality,solving the inversion problem is transformed to minimizing a cost functional,and existence of a minimum is proved by the weakly lower semi-continuity of the functional.Furthermore,the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions,and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.

Approach for Obtaining Material Mechanical Properties in Local Region of Structure Based on Accurate Analysis of Micro-indentation Test

The hot or cold processing would induce the change and the inhomogeneous of the material mechanical properties in the local processing region of the structure,and it is difficult to obtain the specific mechanical properties in these regions by using the traditional material tensile test.To accurately get actual material mechanical properties in the local region of structure,a micro-indentation test system incorporated by an electronic universal material test device has been established.An indenter displacement sensor and a group of special micro-indenter assemblies are estab-lished.A numerical indentation inversion analysis method by using ABAQUS software is also proposed in this study.Based on the above test system and analysis platform,an approach to obtaining material mechanical properties in the local region of structures is proposed and established.The ball indentation test is performed and combined with the energy method by using various changed mechanical properties of 316L austenitic stainless steel under differ-ent elongations.The investigated results indicate that the material mechanical properties and the micro-indentation morphological changes have evidently relevance.Compared with the tensile test results,the deviations of material mechanical parameters,such as hardness H,the hardening exponent n,the yield strength σy and others are within 5%obtained through the indentation test and the finite element analysis.It provides an effective and convenient method for obtaining the actual material mechanical properties in the local processing region of the structure.

Inter-well interferences and their influencing factors during water flooding in fractured-vuggy carbonate reservoirs

Based on the characteristics of injection-production units in fractured-vuggy carbonate reservoirs,nine groups of experiments were designed and performed to analyze the interference characteristics and their influencing factors during water flooding.Based on percolation theory,an inversion model for simulating waterflooding interferences was proposed to study the influence laws of different factors on interference characteristics.The results show that well spacing,permeability ratio,cave size,and cave location all affect the interference characteristics of water flooding.When the cave is located in high permeability fractures,or in the small well spacing direction,or close to the producer in an injection-production unit,the effects of water flooding are much better.When the large cave is located in the high-permeability or small well spacing direction,the well in the direction with lower permeability or smaller well spacing will see water breakthrough earlier.When the cave is in the higher permeability direction and the reserves between the water injector and producer differ greatly,the conductivity differences in different injection-production directions are favorable for water flooding.When the injection-production well pattern is constructed or recombined,it’s better to make the reserves of caves in different injection-production directions proportional to permeability,and inversely proportional to the well spacing.The well close to the cave should be a producer,and the well far from the cave should be an injector.Different ratios of cave reserves to fracture reserves correspond to different optimal well spacings and optimal permeability ratios.Moreover,both optimal well spacing and optimal permeability ratio increase as the ratio of cave reserves to fracture reserves increases.

DYNAMIC STRESS INTENSITY FACTORS AROUND TWO CRACKS NEAR AN INTERFACE OF TWO DISSIMILAR ELASTIC HALF－PLANES UNDER IN－PLANE SHEAR IMPACT LOAD

Transient stresses around two collinear cracks which lie in parallel with theinterface of the two dissimilar half-planes are studied in this article.The surfaces ofthe cracks are sheared suddenly. Application of the Fourier and Laplace transforms technique reduces the problem to that of solving dual integrai equations.To solvethese,the differences of.the crack surface displacements are expanded in a series offunctions which are automatically zero outside of the cracks. The unknown coefficients accompanied in the series are determined by the Schmidt method. The stress intensity .factors are defined in the Laplace transform domain and these are inverted numerically in the physical space .As an example ,the dynamic stress intensity factors around two cracks in a ceramic and steel bonded composite are numerically calculated.

Physical simulation experiment and numerical inversion of the full life cycle of shale gas well

The ultra-low porosity and permeability,as well as complex occurrence and transport state of shale reservoir make it possess special L-type production characteristic curve and complicated shale gas flow mechanism.To solve the difficulty of collecting complete production data due to short production time and operation discontinuity,a full-diameter core physical simulation experiment on the full lifecycle production process of shale gas well depletion is conducted with the purpose of obtaining many important production data including complete pressure and daily gas output in the simulated production process of shale gas well.The experimental results show the production characteristic from simulation is consistent with those from gas well.Based on the simulation data,the critical desorption pressure(12 MPa)of core,free gas production(3820.8 mL),adsorbed gas production(2151.2 mL),the proportion of the daily gas production between free and absorbed gas under different time and formation pressure,as well as the production time and final recovery rate corresponding to abandoned pressure,can be determined accurately.Numerical inversion is carried out to calculate the production performance curve of shale gas well and predict the development effect of gas well based on well testing and similarity analysis of the dimensionless time between core experiment and gas well production.Finally,the permeability and the fracturing effect(fracture network density)as the keys to the effective development of shale gas reservoirs are proposed.The permeability is the fundamental factor and the fracturing technology is the major means.

Estimation of the water–oil–gas relative permeability curve from immiscible WAG coreflood experiments using the cubic B-spline model

Immiscible water-alternating-gas（WAG） flooding is an EOR technique that has proven successful for water drive reservoirs due to its ability to improve displacement and sweep efficiency.Nevertheless,considering the complicated phase behavior and various multiphase flow characteristics,gas tends to break through early in production wells in heterogeneous formations because of overriding,fingering,and channeling,which may result in unfavorable recovery performance.On the basis of phase behavior studies,minimum miscibility pressure measurements,and immiscible WAG coreflood experiments,the cubic B-spline model（CBM） was employed to describe the three-phase relative permeability curve.Using the Levenberg-Marquardt algorithm to adjust the vector of unknown model parameters of the CBM sequentially,optimization of production performance including pressure drop,water cut,and the cumulative gas-oil ratio was performed.A novel numerical inversion method was established for estimation of the water-oil-gas relative permeability curve during the immiscible WAG process.Based on the quantitative characterization of major recovery mechanisms,the proposed method was validated by interpreting coreflood data of the immiscible WAG experiment.The proposed method is reliable and can meet engineering requirements.It provides a basic calculation theory for implicit estimation of oil-water-gas relative permeability curve.

Experimental and numerical studies on the sluggish diffusion in face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys

On purpose of studying the sluggish diffusion of high-entropy alloys, three different face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys were prepared, and assembled into three groups of sandwich- type diffusion multiple annealed at 1273, 1323, and 1373 K respectively. By means of the electron probe microanalyzer technique and recently developed numerical inverse method, the composition- dependent interdiffusivities at different temperatures were effectively evaluated by minimizing the residual between the model-predicted compositions/interdiffusion fluxes and the respectively experi- mental ones. After that, the tracer diffusivities were predicted based on the assessed mobility parameters and thermodynamic descriptions with the simplified ideal solution model. The comprehensive compari- son between the interdiffusivities/tracer diffusivities in the Co-Cr-Cu-Fe-Ni high-entropy alloys and those in sub-binary, ternary, quaternary and other quinary alloys indicates that the sluggish diffusion exists in interdiffusion instead of tracer diffusion for the present Co-Cr-Cu-Fe-Ni high-entropy alloys.

INVERTING GENERATING FUNCTIONS WITH INCREASED NUMERICAL PRECISION-A COMPUTATIONAL EXPERIENCE

In this paper, we consider the numerical inversion of a variety of generating functions （GFs） that arise in the area of engineering and non-engineering fields. Three classes of GFs are taken into account in a comprehensive manner： classes of probability generating functions （PGFs） that are given in rational and non-rational forms, and a class of GFs that are not PGFs. Among others, those PGFs that are not explicitly given but contain a number of unknowns are largely considered as they are often encountered in many interesting applied problems. For the numerical inversion of GFs, we use the methods of the discrete （fast） Fourier transform and the Taylor series expansion. Through these methods, we show that it is remarkably easy to obtain the desired sequence to any given accuracy, so long as enough numerical precision is used in computations. Since high precision is readily available in current software packages and programming languages, one can now lift, with little effort, the so-called Laplacian curtain that veils the sequence of interest. To demonstrate, we take a series of representative examples： the PGF of the number of customers in the discrete-time Geo^X/Geo/c queue, the same in the continuous-time M^X/D/c queue, and the GFs arising in the discrete-time renewal process.

Efficient Numerical Valuation of Continuous Installment Options

In this work we investigate the novel Kryzhnyi method for the numerical inverse Laplace transformation and apply it to the pricing problem of continuous installment options.We compare the results with the one obtained using other classical methods for the inverse Laplace transformation,like the Euler summation method or the Gaver-Stehfest method.

Diffusion growth of Φ ternary intermetallic compound in the Mg-Al-Zn alloy system:In-situ observation and modeling

Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property.However,there is a very lack of existing data due to difficulties in both experimental and computational aspects.The current work aims at presenting the experimental observation on the diffusion growth behavior of Φ phase at 360℃ as well as calculating its composition-dependent interdiffu sion coefficients.We designed and succes s fully fabricated four Mg-τ ternary diffusion couples annealed at 360℃ for different times,where the diffusion path goes across the Φ phase region and the diffusion growth of ternary intermetallic compound can be solely detected.In-situ observation of the time-dependent growth of Φ phase was performed to accurately determine the parabolic growth constant.The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds,which can reproduce the presently observed diffusion growth behavior of Φ ternary intermetallic compound in Mg-τ diffusion couples.