scholarly journals Quadrature Integration Techniques for Random Hyperbolic PDE Problems

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 160
Author(s):  
Rafael Company ◽  
Vera N. Egorova ◽  
Lucas Jódar
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Quadrature Rule ◽  
Inverse Laplace Transform ◽  
Process Time ◽  
Efficient Computation ◽  
Hyperbolic Pde ◽  
Numerical Convergence ◽  
Laplace Transform Technique ◽  
Integration Techniques

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.

Unsteady Hydromagnetic Radiative Flow of a Dusty Fluid Past a Porous Plate With Ramped Wall Temperature

2013 ◽  
Author(s):  
R. Nandkeolyar ◽  
P. Sibanda ◽  
Md. S. Ansari
Keyword(s):  
Magnetic Field ◽  
Laplace Transform ◽  
Porous Plate ◽  
Inverse Laplace Transform ◽  
Combined Effects ◽  
Dusty Fluid ◽  
Electrically Conducting ◽  
Flow Parameters ◽  
Energy Transfers ◽  
Laplace Transform Technique

The combined effects of applied magnetic field, thermal radiation and suction on the flow and free convective heat transfer of a viscous, incompressible, electrically conducting dusty fluid past a flat plate with ramped temperature are studied. The governing partial differential equations for momentum and energy transfers, for both the fluid and particle phases, are solved using Laplace transform technique. The inverse Laplace transform is obtained numerically using Matlab. A comparison of Numerical solution and analytical solution for energy transfer is made which shows an excellent agreement. The effects of pertinent flow parameters are analyzed with the help of graphs and tables.

scholarly journals Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xiaoli Qiang ◽  
Kamran ◽  
Abid Mahboob ◽  
Yu-Ming Chu
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Fractional Order ◽  
Integrodifferential Equations ◽  
Inverse Laplace Transform ◽  
Laplace Domain ◽  
The Real ◽  
Volterra Integrodifferential Equation ◽  
Engineering Sciences ◽  
Real Domain

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

scholarly journals Solution for the Multigroup Neutron Space Kinetics Equations by Source Iterative Method

2021 ◽  
Vol 9 (2A) ◽  
Author(s):  
Matheus Gularte Tavares
Keyword(s):  
Laplace Transform ◽  
Neutron Flux ◽  
Source Term ◽  
Polynomial Interpolation ◽  
Initial Guess ◽  
Inverse Laplace Transform ◽  
Numerical Inverse Laplace Transform ◽  
Laplace Transform Technique ◽  
Linear Differential ◽  
Multigroup Neutron

In this work, we used a modified Picard’s method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a first order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform by Stehfest method. We present numerical simulations and comparisons with available results in literature.

scholarly journals Resonance of Nanoscale Beam due to Various Sources in Modified Couple Stress Thermoelastic Diffusion with Phase Lags

2019 ◽  
Vol 23 (1) ◽  
pp. 36-49
Author(s):  
Rajneesh Kumar ◽  
Shaloo Devi ◽  
Veena Sharma
Keyword(s):  
Laplace Transform ◽  
Couple Stress ◽  
Chemical Potential ◽  
Inverse Laplace Transform ◽  
Phase Lag ◽  
Thermoelastic Diffusion ◽  
Special Cases ◽  
Laplace Transform Technique ◽  
Modified Couple Stress ◽  
Phase Lags

Abstract This paper deals with the study of thermoelastic thin beam in a modified couple stress with three-phase-lag thermoelastic diffusion model subjected to thermal and chemical potential sources. The governing equations are derived by using the Euler-Bernoulli beam assumption and eigenvalue approach. The Laplace transform technique is employed to obtain the expressions for displacements, lateral deflection, temperature change, axial stress and chemical potential. A particular type of instantaneous and distributed sources is taken to show the utility of the approach. The general algorithm of the inverse Laplace transform is developed to compute the results numerically. The numerical results are depicted graphically to show the effects of phase lags, with and without energy dissipation on the resulting quantities. Some special cases are given.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Modeling electro-osmotic flow and thermal transport of Caputo fractional Burgers fluid through a micro-channel

2021 ◽  
pp. 095440892110259
Author(s):  
Mohammed Abdulhameed ◽  
Garba Tahiru Adamu ◽  
Gulibur Yakubu Dauda
Keyword(s):  
Electric Field ◽  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Micro Channel ◽  
Osmotic Flow ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Burgers Fluid ◽  
Electro Osmotic Flow

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

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