Analytical solution of the one-dimensional nonlinear Richards equation based on special hydraulic functions and the variational principle

2018 ◽  
Vol 69 (6) ◽  
pp. 980-996
Author(s):  
L. Su ◽  
Q. Wang ◽  
X. Qin ◽  
Y. Shan ◽  
B. Zhou ◽  
...  
Keyword(s):  
Variational Principle ◽  
Analytical Solution ◽  
Richards Equation ◽  
One Dimensional ◽  
The One

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.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

An Exact Solution to Steady Heat Conduction in a Two-Dimensional Annulus on a One-Dimensional Fin: Application to Frosted Heat Exchangers With Round Tubes

2005 ◽  
Vol 128 (4) ◽  
pp. 397-404 ◽  
Author(s):  
A. D. Sommers ◽  
A. M. Jacobi
Keyword(s):  
Thermal Conductivity ◽  
Analytical Solution ◽  
High Thermal Conductivity ◽  
Two Dimensional ◽  
One Dimensional ◽  
Low Thermal Conductivity ◽  
Fin Efficiency ◽  
Coated Metal ◽  
The One ◽  
Term Approximation

The fin efficiency of a high-thermal-conductivity substrate coated with a low-thermal-conductivity layer is considered, and an analytical solution is presented and compared to alternative approaches for calculating fin efficiency. This model is appropriate for frost formation on a round-tube-and-fin metallic heat exchanger, and the problem can be cast as conduction in a composite two-dimensional circular cylinder on a one-dimensional radial fin. The analytical solution gives rise to an eigenvalue problem with an unusual orthogonality condition. A one-term approximation to this new analytical solution provides fin efficiency calculations of engineering accuracy for a range of conditions, including most frosted-coated metal fins. The series solution and the one-term approximation are of sufficient generality to be useful for other cases of a low-thermal-conductivity coating on a high-thermal-conductivity substrate.

LOCALIZED BREATHER-LIKE EXCITATIONS IN THE HELICOIDAL PEYRARD–BISHOP MODEL OF DNA

2009 ◽  
Vol 02 (04) ◽  
pp. 405-417 ◽  
Author(s):  
CONRAD BERTRAND TABI ◽  
ALIDOU MOHAMADOU ◽  
TIMOLEON CREPIN KOFANE
Keyword(s):  
Analytical Solution ◽  
Continuous Wave ◽  
Exact Analytical Solution ◽  
Modulational Instability ◽  
Elliptic Functions ◽  
Instability Criterion ◽  
Dna Dynamics ◽  
Jacobian Elliptic Functions ◽  
One Dimensional ◽  
The One

We consider the one-dimensional helicoidal Peyrard–Bishop (PB) model of DNA dynamics. By means of a method based on the Jacobian elliptic functions, we obtain the exact analytical solution which describes the modulational instability and the propagation of a bright solitary wave on a continuous wave background. It is shown that these solutions depend on the modulational (or Benjamin-Feir) instability criterion. Numerical simulations of their propagation show these excitations to be long-lived and suggest that they are physically relevant for DNA.

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.

Thermochemical Response of Honeycomb Sandwich Panels

1983 ◽  
Vol 1 (5) ◽  
pp. 379-395
Author(s):  
Kumar Ramohalli
Keyword(s):  
Thermal Conductivity ◽  
Analytical Solution ◽  
Thermal Conductivity Coefficient ◽  
Sandwich Panels ◽  
One Dimensional ◽  
Dimensional Approximation ◽  
Honeycomb Sandwich ◽  
The One

A simple study aimed at predicting the Thermochemical Response of honey comb sandwich panels is presented. The overall thermal conductivity coefficient for the panel is obtained through a consideration of the convective gas move ment within the cell spaces. The earlier correlations of Catton and Edwards are used. The analytical solution for the one-dimensional approximation is quoted from an earlier study.

scholarly journals Analytical assessment of subsidence due to Aquifer Storage and Recovery (ASR) applications to multi-aquifer systems

2000 ◽  
Vol 21 ◽  
Author(s):  
J. Li
Keyword(s):  
Analytical Solution ◽  
Land Subsidence ◽  
Potential Risk ◽  
Aquifer Storage And Recovery ◽  
One Dimensional ◽  
Aquifer System ◽  
Aquifer Storage ◽  
Aquifer Systems ◽  
The One ◽  
Development And Management

The present paper emphasises concerns of land subsidence or compression of clay confining beds caused by periodic withdrawal and injection of water from or into the adjacent aquifers. An analytical solution for a one-dimensional case based on a sandwich model is found so that analysis of potential risk of aquifer system deformation due to the technology of Aquifer Storage and Recovery (ASR) can be conducted. A governing equation expressed directly in terms of displacement is employed to describe the one-dimensional subsidence. For simplicity, saturated aquifer systems are assumed to behave like poroelastic material. A cyclic loading function with a triangle pattern is assumed at boundaries to simulate effective stress induced by changes in hydraulic head at boundaries. The both compression and swelling of clay due to the periodic and linear loads at the boundaries are considered in this model. The two aquifers (one above the confining bed and the other beneath) can be pumped independently of each other. The results from the analytical solution are applied to estimate and predict potential risk of land subsidence due to ASR activity and to provide a first-estimate type of guideline for city or regional development and management of water resources.

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