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 sum

New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology

We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential-difference equations. The proposed method is based on the Laplace trans- form with the homotopy analysis method （HAM）. This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Mechanisms of Proton-Proton Inelastic Cross-Section Growth in Multi-Peripheral Model within the Framework of Perturbation Theory.Part 3

We develop a new method for taking into account the interference contributions to proton-proton inelastic cross-section within the framework of the simplest multi-peripheral model based on the self-interacting scalar φ3 field theory, using Laplace’s method for calculation of each interference contribution. We do not know any works that adopted the inter- ference contributions for inelastic processes. This is due to the generally adopted assumption that the main contribution to the integrals expressing the cross section makes multi-Regge domains with its characteristic strong ordering of secon- dary particles by rapidity. However, in this work, we find what kind of space domains makes a major contribution to the integral and these space domains are not multi-Regge. We demonstrated that because these interference contributions are significant, so they cannot be limited by a small part of them. With the help of the approximate replacement the sum of a huge number of these contributions by the integral were calculated partial cross sections for such numbers of secondary particles for which direct calculation would be impossible. The offered model qualitative agrees with experimental dependence of total scattering cross-section on energy with a characteristic minimum in the range ≈ 10 GeV. However, quantitative agreement was not achieved;we assume that due to the fact that we have examined the simplest diagrams of theory.

New Fuzzy Fractional Epidemic Model Involving Death Population

In this research,we propose a new change in classical epidemic models by including the change in the rate of death in the overall population.The existing models like Susceptible-Infected-Recovered(SIR)and Susceptible-Infected-Recovered-Susceptible(SIRS)include the death rate as one of the parameters to estimate the change in susceptible,infected and recovered populations.Actually,because of the deficiencies in immunity,even the ordinary flu could cause death.If people’s disease resistance is strong,then serious diseases may not result in mortalities.The classical model always assumes a closed system where there is no new birth or death,no immigration or emigration,while in reality,such assumptions are not realistic.Moreover,the classical epidemic model does not report the change in population due to death caused by a disease.With this study,we try to incorporate the rate of change in the population of death caused by a disease,where the model is framed to reduce the curve of death along with the susceptible and infected populations.Since the rate of change turned out to be very small,we have tried to estimate it fractionally.Thus,the model is defined using fuzzy logic and is solved by two different methods:a Laplace Adomian decomposition method(LADM)and a differential transform method(DTM)for an arbitrary order α.To test its accuracy,we compared the results of both DTM and LADM with the fourth-order Runge-Kutta method(RKM-4)at α=1.

Numerical Analysis and Transformative Predictions of Fractional Order Epidemic Model during COVID-19 Pandemic: A Critical Study from Bangladesh

The COVID-19 pandemic is a curse and a threat to global health, development, the economy, and peaceful society because of its massive transmission and high rates of mutation. More than 220 countries have been affected by COVID-19. The world is now facing a drastic situation because of this ongoing virus. Bangladesh is also dealing with this issue, and due to its dense population, it is particularly vulnerable to the spread of COVID-19. Recently, many non-linear systems have been proposed to solve the SIR (Susceptible, Infected, and Recovered) model for predicting Coronavirus cases. In this paper, we have discussed the fractional order SIR epidemic model of a non-fatal disease in a population of a constant size. Using the Laplace Adomian Decomposition method, we get an approximate solution to the model. To predict the dynamic transmission of COVID-19 in Bangladesh, we provide a numerical argument based on real data. We also conducted a comparative analysis among susceptible, infected, and recovered people. Furthermore, the most sensitive parameters for the Basic Reproduction Number (R0) are graphically presented, and the impact of the compartments on the transmission dynamics of the COVID-19 pandemic is thoroughly investigated.

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.

The bound state solution for the Morse potential with a localized mass profile

We investigate an analytical solution for the Schr o¨dinger equation with a position-dependent mass distribution, with the Morse potential via Laplace transformations. We considered a mass function localized around the equilibrium position.The mass distribution depends on the energy spectrum of the state and the intrinsic parameters of the Morse potential. An exact bound state solution is obtained in the presence of this mass distribution.

Analytical solution for a circular roadway considering the transient effect of excavation unloading

The rocks surrounding a roadway exhibit some special and complex phenomena with increasing depth of excavation in underground engineering.Quasi-static analysis cannot adequately explain these engineering problems.The computational model of a circular roadway considering the transient effect of excavation unloading is established for these problems.The time factor makes the solution of the problem difficult.Thus,the computational model is divided into a dynamic model and a static model.The Laplace integral transform and inverse transform are performed to solve the dynamic model and elasticity theory is used to analyze the static model.The results from an example show that circumferential stress increases and radial stress decreases with time.The stress difference becomes large gradually in this progress.The displacement increases with unloading time and decreases with the radial depth of surrounding rocks.It can be seen that the development trend of unloading and displacement is similar by comparing their rates.Finally,the results of ANSYS are used to verify the analytical solution.The contrast indicates that the laws of the two methods are basically in agreement.Thus,the analysis can provide a reference for further study.

Mathematical analysis of Hepatitis C Virus infection model in the framework of non-local and non-singular kernel fractionalderivative

In this paper,we study a mathematical model of Hepatitis C Virus(HCV)infection.We present a compartmental mathematical model involving healthy hepatocytes,infected hepatocytes,non-activated dendritic cells,activated dendritic cells and cytotoxic T lymphocytes.The derivative used is of non-local fractional order and with non-singular kernel.The existence and uniqueness of the system is proven and its stability is analyzed.Then,by applying the Laplace Adomian decomposition method for the fractional derivative,we present the semi-analytical solution of the model.Finally,some numerical simulations are performed for concrete values of the parameters and several graphs are plotted to reveal the qualitative properties of the solutions.

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors,vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration.However,traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function,so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation.In this paper,the improved Gaver-Stehfest(G-S)inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration,and the optimized G-S coefficients are given.The effectiveness of the algorithm is verified by numerical examples.Compared with the linear transformation accelerated convergence algorithm,it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform,and the time consumption is relatively stable and short,which provides a reliable calculation method for the study of electromagnetic mechanics,wave propagation,and fracture dynamics.