scholarly journals Another Approach to the Extended Stokes’ Problems for the Oldroyd-B Fluid

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Chi-Min Liu
Keyword(s):  
Boundary Condition ◽  
Laplace Transform ◽  
Series Expansion ◽  
Initial Conditions ◽  
Step Function ◽  
Inverse Laplace Transform ◽  
Mathematical Methods ◽  
Unit Step ◽  
Stokes Problems ◽  
Heaviside Unit Step Function

The extended Stokes problems, which study the flow suddenly driven by relatively moving half-planes, are reexamined for the Oldroyd-B fluid. This topic has been studied (Liu, 2011) by applying the series expansion to calculate the inverse Laplace transform. The derived solution was correct but tough to perform the calculation due to the series expansion of infinite terms. Herein another approach, the contour integration, is applied to calculate the inversion. Moreover, the Heaviside unit step function is included into the boundary condition to ensure the consistence between boundary and initial conditions. Mathematical methods used herein can be applied to other fluids for the extended Stokes’ problems.

A Rotary Magnetic Damper Consisting of Several Sector Magnets and an Arbitrarily Shaped Conductor With a Circular Cavity

1986 ◽  
Vol 108 (4) ◽  
pp. 314-321
Author(s):  
Yasuo Karube ◽  
Kosuke Nagaya
Keyword(s):  
Boundary Condition ◽  
Electromagnetic Field ◽  
Magnetic Flux ◽  
Differential Equations ◽  
Fourier Expansion ◽  
Step Function ◽  
Circular Cavity ◽  
Damping Force ◽  
Damping Coefficients ◽  
Unit Step

In this paper, the damping force and the damping coefficient of a rotary magnetic damper consisting of several sector magnets and an arbitrarily shaped plate conductor with a circular cavity have been obtained theoretically. The unit step function is applied to solve the differential equations of the electromagnetic field, and the boundary condition of the outer arbitrarily shaped boundary of the plate conductor is satisfied directly by making use of the Fourier expansion collocation method. Numerical calculations have been carried out for the dimensionless damping coefficients with the variations of various factors such as the magnetic flux range, the outer shape and the radius of the inner circular cavity of the conductor, the position and the number of the magnets.

A General Time Domain Solution for the Waterhammer of Liquid-Filled Piping System

2011 ◽  
Vol 243-249 ◽  
pp. 4488-4495 ◽  
Author(s):  
Ke Yang
Keyword(s):  
Numerical Calculation ◽  
Laplace Transform ◽  
Time Domain ◽  
Initial Conditions ◽  
Wave Functions ◽  
Inverse Laplace Transform ◽  
Piping System ◽  
Algebraic Equations ◽  
Final Solution ◽  
General Time

The paper has obtained a unified final solution for the waterhammer equations. The proposed solution, covering all kinds of initial conditions and boundary conditions, has been proved to be written in the form of the d'Alembert's wave functions. The periodical influence of the initial conditions on the results is discussed. The proposed solution, with two kinds of algebraic equations containing only finite terms, is suitable for numerical calculation, convenient for programming and liable to dealing with complex pipe systems. An example has been given to show the use of the method. The skill to perform the inverse Laplace transform in obtaining the solution is different from the traditional ones and can be extended to use in many other problems including the FSI waterhammer problem.

The Laplace transform and spectral representations of polynomial operator pencils

1993 ◽  
Vol 443 (1918) ◽  
pp. 333-342
Keyword(s):  
Laplace Transform ◽  
Spectral Representation ◽  
Eigenvalue Problems ◽  
Initial Conditions ◽  
Classical Problem ◽  
Operator Pencil ◽  
Linear Operators ◽  
Inverse Laplace Transform ◽  
Polynomial Operator ◽  
The Laplace Transform

The spectral representation associated with the polynomial operator pencil L 0 + λL 1 + λ 2 L 2 +. . . λ N L N , where L n ( n = 0,1,2,..., N ) are linear operators and λ is a complex parameter, is derived formally using the Laplace transform. The derivation involves a conversion of the eigenvalue problem for the operator pencil into an initial-value problem by replacing λ with ∂/∂t and introducing N -1 initial conditions. This procedure yields the spectral representation in the form of an inverse Laplace transform of the Green’s operator associated with the operator pencil. The results of this paper are illustrated with examples and provide a simple but powerful and systematic approach to non-standard eigenvalue problems for linear operators. These examples are a 2 x 2 matrix problem which has three eigenvalues, a Sturm-Liouville-Rossby type wave equation discussed recently by A. Masuda ( Q. appl. Math . 47, 435-445 (1989)), and a classical problem in which the eigenvalue parameter appears not only in the differential equation but also in the boundary conditions.

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 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.

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.

MACSYMS's inverse Laplace transform

1989 ◽  
Vol 23 (1) ◽  
pp. 33-38
Author(s):  
M. Clarkson
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform

Numerical Inversion of Laplace Transform for Time Resolved Thermal Characterization Experiment

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
J. Toutain ◽  
J.-L. Battaglia ◽  
C. Pradere ◽  
J. Pailhes ◽  
A. Kusiak ◽  
...  
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Periodic Function ◽  
Thermal Characterization ◽  
Inverse Laplace Transform ◽  
Numerical Inversion ◽  
Time Resolved ◽  
Reliable Technique ◽  
Numerical Inverse Laplace Transform ◽  
Thermal Disturbance

The aim of this technical brief is to test numerical inverse Laplace transform methods with application in the framework of the thermal characterization experiment. The objective is to find the most reliable technique in the case of a time resolved experiment based on a thermal disturbance in the form of a periodic function or a distribution. The reliability of methods based on the Fourier series methods is demonstrated.

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