An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hypergeometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

An Analytical Solution of the Generalized Phase-Lagging Equation for Ultra-Fast Heat Transfer in One-Dimensional Semi-Infinite Domain

2012 ◽  
Author(s):  
Vladimir Kulish ◽  
Kirill V. Poletkin
Keyword(s):  
Analytical Solution ◽  
Domain Boundary ◽  
Integral Form ◽  
Solution Procedure ◽  
Generalized Derivatives ◽  
Infinite Domain ◽  
One Dimensional ◽  
Novel Technique ◽  
The Laplace Transform ◽  
Derivatives Of

The paper presents an integral solution of the generalized one-dimensional phase-lagging heat equation with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the temperature and heat flux. The solution is valid everywhere within the domain, including the domain boundary.

Analytical Solution of 1-D Viscoelastic Consolidation of Soft Soils with Fractional Maxwell Model

2012 ◽  
Vol 157-158 ◽  
pp. 419-423
Author(s):  
Ya Peng Zhang ◽  
Feng Gao
Keyword(s):  
Analytical Solution ◽  
Soft Soil ◽  
Soil Layer ◽  
Viscoelastic Model ◽  
Integral Form ◽  
Maxwell Model ◽  
One Dimensional ◽  
The Laplace Transform ◽  
Fractional Maxwell Model ◽  
The One

Considering the rheological characteristics of soil, think the fractional maxwell with viscoelastic model can be described, the fractional maxwell model into integral form of saturated soft soil layer, the one dimensional compression, through the Laplace transform problems get instantaneous loading and single stage, the analytical solution of the loading conditions.

scholarly journals Thermoelastic Diffusion Multicomponent Half-Space under the Effect of Surface and Bulk Unsteady Perturbations

2019 ◽  
Vol 24 (1) ◽  
pp. 26 ◽  
Author(s):  
Sergey Davydov ◽  
Andrei Zemskov ◽  
Elena Akhmetova
Keyword(s):  
Half Space ◽  
Green's Functions ◽  
Green’S Functions ◽  
Local Equilibrium ◽  
Linear Equations ◽  
Integral Form ◽  
External Effects ◽  
Thermoelastic Diffusion ◽  
One Dimensional ◽  
The Laplace Transform

This article presents an algorithm for solving the unsteady problem of one-dimensional coupled thermoelastic diffusion perturbations propagation in a multicomponent isotropic half-space, as a result of surface and bulk external effects. One-dimensional physico-mechanical processes, in a continuum, have been described by a local-equilibrium model, which included the coupled linear equations of an elastic medium motion, heat transfer, and mass transfer. The unknown functions of displacement, temperature, and concentration increments were sought in the integral form, which was a convolution of the surface and bulk Green’s functions and external effects functions. The Laplace transform on time and the Fourier sine and cosine transforms on the coordinate were used to find the Green’s functions. The obtained Green’s functions was analyzed. Test calculations were performed on the examples of some technological processes.

scholarly journals The Analytical Solution of Parabolic Volterra Integro- Differential Equations in the Infinite Domain

2016 ◽  
Author(s):  
Yun Zhao ◽  
Feng-Qun Zhao
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Kernel Functions ◽  
Infinite Domain ◽  
Memory Kernel ◽  
One Dimensional ◽  
Different Types ◽  
Frictional Memory Kernel

This article focuses on obtaining the analytical solutions for parabolic Volterra integro- differential equations in d-dimensional with different types frictional memory kernel. Based on theories of Laplace transform, Fourier transform, the properties of Fox-H function and convolution theorem, analytical solutions of the equations in the infinite domain are derived under three frictional memory kernel functions respectively. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, Fox-H function and convolution form of Fourier transform. In addition, the graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equation with power-law memory kernel. It can be seen that the solution curves subject to Gaussian decay at any given moment.

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 Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

The combined effect of lattice’s uniaxial strains and electron–phonon interaction’s screening on TBEC of the intersite bipolarons

2016 ◽  
Vol 30 (26) ◽  
pp. 1650186
Author(s):  
B. Yavidov ◽  
SH. Djumanov ◽  
T. Saparbaev ◽  
O. Ganiyev ◽  
S. Zholdassova ◽  
...  
Keyword(s):  
Ideal Gas ◽  
Einstein Condensation ◽  
Chain Model ◽  
One Dimensional ◽  
Electron Phonon Interaction ◽  
Model Lattice ◽  
Experimental Values ◽  
Bose Einstein ◽  
Derivatives Of ◽  
The Ideal

Having accepted a more generalized form for density-displacement type electron–phonon interaction (EPI) force we studied the simultaneous effect of uniaxial strains and EPI’s screening on the temperature of Bose–Einstein condensation [Formula: see text] of the ideal gas of intersite bipolarons. [Formula: see text] of the ideal gas of intersite bipolarons is calculated as a function of both strain and screening radius for a one-dimensional chain model of cuprates within the framework of Extended Holstein–Hubbard model. It is shown that the chain model lattice comprises the essential features of cuprates regarding of strain and screening effects on transition temperature [Formula: see text] of superconductivity. The obtained values of strain derivatives of [Formula: see text] [Formula: see text] are in qualitative agreement with the experimental values of [Formula: see text] [Formula: see text] of La[Formula: see text]Sr[Formula: see text]CuO4 under moderate screening regimes.

Building Vertical Walls by Deposition of Molten Metal Droplets

2005 ◽  
Author(s):  
M. Fang ◽  
S. Chandra ◽  
C. B. Park
Keyword(s):  
Surface Layer ◽  
Analytical Solution ◽  
Substrate Temperature ◽  
Layer By Layer ◽  
Droplet Temperature ◽  
Droplet Generator ◽  
One Dimensional ◽  
Metallurgical Bonding ◽  
Experimental Parameters ◽  
Vertical Walls

Experiments were conducted to determine conditions under which good metallurgical bonding was achieved in vertical walls composed of multiple layers of droplets that were fabricated by depositing tin droplets layer by layer. Molten tin droplets (0.75 mm diameter) were deposited using a pneumatic droplet generator on an aluminum substrate. The primary parameters varied in experiments were those found to most affect bonding between droplets on different layers: droplet temperature (varied from 250°C to 325°C) and substrate temperature (varied from 100°C to 190°C). Considering the cooling rate of droplet is much faster than the deposition rate previous deposition layer cooled down too much that impinging droplets could only remelt a thin surface layer after impact. Assuming that remelting between impacting droplets and the previous deposition layer is a one-dimensional Stefan problem with phase change an analytical solution can be found and applied to predict the minimum droplet temperature and substrate temperature required for local remelting. It was experimentally confirmed that good bonding at the interface of two adjacent layers could be achieved when the experimental parameters were such that the model predicted remelting.

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