On Fuzzy Conformable Double Laplace Transform with Applications to Partial Differential Equations

The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

QUATERNIONIC SLICE REGULAR FUNCTIONS AND QUATERNIONIC LAPLACE TRANSFORMS

The functions studied in the paper are the quaternion-valued functions of a quaternionic variable.It is shown that the left slice regular functions and right slice regular functions are related by a particular involution,and that the intrinsic slice regular functions play a central role in the theory of slice regular functions.The relation between left slice regular functions,right slice regular functions and intrinsic slice regular functions is revealed.As an application,the classical Laplace transform is generalized naturally to quaternions in two different ways,which transform a quaternion-valued function of a real variable to a left or right slice regular function.The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.

LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

Assume that 0<p<∞ and that B is a connected nonempty open set in R^(n),and that A^(p)(B)is the vector space of all holomorphic functions F in the tubular domains R^(n)+iB such that for any compact set K⊂B,‖ y →‖x →F(x+iy)‖Lp(R^(n))‖ L(K)<∞,so A^(p)(B)is a Frechet space with the Heine-Borel property,its topology is induced by a complete invariant metric,is not locally bounded,and hence is not normal.Furthermore,if 1≤p≤2,then the element F of A^(p)(B)can be written as a Laplace transform of some function f∈L(R^(n)).

Contrast Enhancement Using Weighted Coupled Histogram Equalization with Laplace Transform

Histogram equalization is a traditional algorithm improving the image contrast,but it comes at the cost of mean brightness shift and details loss.In order to solve these problems,a novel approach to processing foreground pixels and background pixels independently is proposed and investigated.Since details are mainly contained in the foreground,the weighted coupling of histogram equalization and Laplace transform were adopted to balance contrast enhancement and details preservation.The weighting factors of image foreground and background were determined by the amount of their respective information.The proposed method was conducted to images acquired from CVG⁃UGR and US⁃SIPI image databases and then compared with other methods such as clipping histogram spikes,histogram addition,and non⁃linear transformation to verify its validity.Results show that the proposed algorithm can effectively enhance the contrast without introducing distortions,and preserve the mean brightness and details well at the same time.

On the Laplace transform of delta function

Delta function is an important function in mathematics and physics. In this paper, the Laplace transforms of δ(t)and δ(t-τ)have been discussed in detail. After the Laplace transform of δ(t)is analyzed, the author has found that three aspects should be taken into account, i.e. τ→0+, τ→0- andτ=0; and it is the same with the Laplace transform of δ(t-τ). Then the results of the Laplace transform of Delta function have been obtained in a rigorous and comprehensive sense.

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.

Analysis of Transient Pulse Electroosmotic Flow of Maxwell Fluid through a Circular Micro-Channel Using Laplace Transform Method

A semi-analytical solution is presented using method of Laplace transform for the transient pulse electroosmotic flow (EOF) of Maxwell fluid in a circular micro-channel. The driving mode of pulse EOF here is considered as an ideal rectangle pulse. The solution involves solving the linearized Poisson-Boltzmann (P-B) equation, together with the Cauchy momentum equation and the general Maxwell constitutive equation. The results show that the profiles of pulse EOF velocity vary rapidly and gradually stabilize as the increase of time within a half period. The velocity profiles at the center of the micro-channel increase significantly with relaxation time , especially for the smaller pulse width a. However, as the pulse width a increases, this change will be less obvious. At the same time, the different change frequency of velocity profiles will slow down, which means a long cycle time. Additionally, the time needed to attain the steady status becomes longer with the increase of relaxation time and pulse width a.

Laplace Transform, Non-Constant Coefficients Differential Equations and Applications to Riccati Equation

In this paper, the Laplace Transform is used to find explicit solutions of a fam-ily of second order Differential Equations with non-constant coefficients. For some of these equations, it is possible to find the solutions using standard tech-niques of solving Ordinary Differential Equations. For others, it seems to be very difficult indeed impossible to find explicit solutions using traditional methods. The Laplace transform could be an alternative way. An application on solving a Riccati Equation is given. Recall that the Riccati Equation is a non-linear differential equation that arises in many topics of Quantum Me-chanics and Physics.

A Regularization Method for Approximating the Inverse Laplace Transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite

On THE NUMERICAL INVERSION OF THE LAPLACE TRANSFORM BY THE USE OF AN OPTIMIZED LEGENDRE POLYNOMIALS

A method for inverting the Laplace transform is presented, using a finite series of the classical Legendre polynomials. The method recovers a real valued function f(t) in a finite interval of the positive real axis when f(t) belongs to a certain class βand requires the knowledge of its Laplace transform F(s) only at a finite number of discrete points on the real axis s>0. The choice of these points will be carefully considered so as to improve the approximation error as well as to minimize the number of steps needed in the evaluations. The method is tested on few examples, with particular emphasis on the estimation of the error bounds involved.

q-Laplace Transform

The Fourier transformations are used mainly with respect to the space variables. In certain circumstances, however, for reasons of expedience or necessity, it is desirable to eliminate time as a variable in the problem. This is achieved by means of the Laplace transformation. We specify the particular concepts of the q-Laplace transform. The convolution for these transforms is considered in some detail.

Finite Element Method for Transient Electric Field by Using Indirect Laplace Transform with High Accuracy

This paper focuses on the finite element method in the complex frequency domain(CFD-FEM)for the transient electric field.First,the initial value and boundary value problem of the transient electric field under the electroquasistatic field in the complex frequency domain is given.In addition,the finite element equation and the constrained electric field equation on the boundary are derived.Secondly,the indirect algorithm of the numerical inverse Laplace transform is introduced.Based on it,the calculation procedures of the CFD-FEM are illustrated in detail.Thirdly,the step response,zero-state response under the positive periodic square waveform(PPSW)voltage,and the zero-input response by the CFD-FEM with direct algorithm and indirect algorithm are compared.Finally,the reason for the numerical oscillations of the zero-state response under the PPSW voltage is analyzed,and the method to reduce oscillations is proposed.The results show that the numerical accuracy of the indirect algorithm of the CFD-FEM is more than an order of magnitude higher than that of the direct algorithm when calculating the step response of the transient electric field.The proposed method can significantly reduce the numerical oscillations of the zero-state response under the PPSW voltage.The proposed method is helpful for the calculation of the transient electric field,especially in the case of frequency-dependent parameters.

Calculating the transient behavior of grounding systems using inverse Laplace transform

This paper deals with a unified and novel approach for analyzing the frequency and time domain performance of grounding systems.The proposed procedure is based on solving the full set of Maxwell's equations in the frequency domain,and enables the exact computation of very near fields at the surface of the grounding grid,as well as far fields,by simple and accurate closed-form expressions for solving Sommerfeld integrals.In addition,the soil ionization is easily considered in the proposed method.The frequency domain responses are converted to the time domain by fast inverse Laplace transform.The results are validated and have shown acceptable accuracy.

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials(FGMs).The three methods are,respectively,the method of fundamental solution(MFS),the boundary knot method(BKM),and the collocation Trefftz method(CTM)in conjunction with Kirchhoff transformation and various variable transformations.In the analysis,Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions.The proposed MFS,BKM and CTM are mathematically simple,easyto-programming,meshless,highly accurate and integration-free.Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered,and the results are compared with those from meshless local boundary integral equation method(LBIEM)and analytical solutions to demonstrate the effi-ciency of the present schemes.

Analysis for pressure transient of coalbed methane reservoir based on Laplace transform finite difference method

Based on fractal geometry,fractal medium of coalbed methane mathematical model is established by Langmuir isotherm adsorption formula,Fick's diffusion law,Laplace transform formula,considering the well bore storage effect and skin effect.The Laplace transform finite difference method is used to solve the mathematical model.With Stehfest numerical inversion,the distribution of dimensionless well bore flowing pressure and its derivative was obtained in real space.According to compare with the results from the analytical method,the result from Laplace transform finite difference method turns out to be accurate.The influence factors are analyzed,including fractal dimension,fractal index,skin factor,well bore storage coefficient,energy storage ratio,interporosity flow coefficient and the adsorption factor.The calculating error of Laplace transform difference method is small.Laplace transform difference method has advantages in well-test application since any moment simulation does not rely on other moment results and space grid.

Fast Laplace transform methods for the PDE system of Parisian and Parasian option pricing

This paper develops a fast Laplace transform method for solving the complex PDE system arising from Parisian and Parasian option pricing.The value functions of the options are governed by a system of partial differential equations(PDEs)of two and three dimensions.Applying the Laplace transform to the PDEs with respect to the calendar time to maturity leads to a coupled system consisting of an ordinary differential equation(ODE)and a 2-dimensional partial differential equation(2d-PDE).The solution to this ODE is found analytically on a specific parabola contour that is used in the fast Laplace inversion,whereas the solution to the 2d-PDE is approximated by solving 1-dimensional integro-differential equations.The Laplace inversion is realized by the fast contour integral methods.Numerical results confirm that the Laplace transform methods have the exponential convergence rates and are more efficient than the implicit finite difference methods,Monte Carlo methods and moving window methods.

Solution of the Schrodinger equation for a particular form of Morse potential using the Laplace transform

In this paper, we have solved the Schrdinger equation for a particular kind of Morse potential and find its normalized eigenfunctions and eigenvalues, exactly. Our work is based on the Laplace transform technique which reduces the second-order differential equation to a first-order.

Numerical inversion method for the Laplace transform based on Boubaker polynomials operational matrix

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix.The efficiency of the presented approach is demonstrated by solving some differential equations.Also,this technique is combined with the standard Laplace Homotopy Per-turbation Method.The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.

A Hybrid Finite Element-Laplace Transform Method for the Analysis of Transient Electromagnetic Scattering by an Over-Filled Cavity in the Ground Plane

A hybrid finite element-Laplace transform method is implemented to analyze the time domain electromagnetic scattering induced by a 2-D overfilled cavity embedded in the infinite ground plane.The algorithm divides the whole scattering domain into two,interior and exterior,sub-domains.In the interior sub-domain which covers the cavity,the problem is solved via the finite element method.The problem is solved analytically in the exterior sub-domain which slightly overlaps the interior subdomain and extends to the rest of the upper half plane.The use of the Laplace transform leads to an analytical link condition between the overlapping sub-domains.The analytical link guides the selection of the overlapping zone and eliminates the need to use the conventional Schwartz iteration.This dramatically improves the efficiency for solving transient scattering problems.Numerical solutions are tested favorably against analytical ones for a canonical geometry.The perfect link over the artificial boundary between the finite element approximation in the interior and analytical solution in the exterior further indicates the reliability of the method.An error analysis is also performed.