Eigenvalue Expansion Approach to Study Bio-Heat Equation

2016 ◽  
Vol 07 (02) ◽  
pp. 1650002
Author(s):  
M. A. Khanday ◽  
Khalid Nazir
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Metabolic Pathways ◽  
Biological Tissues ◽  
Temperature Profiles ◽  
Order Partial Differential Equation ◽  
Eigenvalue Method ◽  
Thermal Disturbance ◽  
Heat Processes ◽  
Eigenvalue Expansion

A mathematical model based on Pennes bio-heat equation was formulated to estimate temperature profiles at peripheral regions of human body. The heat processes due to diffusion, perfusion and metabolic pathways were considered to establish the second-order partial differential equation together with initial and boundary conditions. The model was solved using eigenvalue method and the numerical values of the physiological parameters were used to understand the thermal disturbance on the biological tissues. The results were illustrated at atmospheric temperatures [Formula: see text]C and [Formula: see text]C.

scholarly journals Derivation of Second Order Partial Differential Equation Indicating Wave and Heat Equation through the Use of the Navier Stoke’s Equation for Unsteady and Incompressible Flow

2021 ◽  
Vol 2 (3) ◽  
pp. 19-21
Author(s):  
M. G. A. Hayder Chowdhury ◽  
N. Akhtar
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Incompressible Flow ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Strong Relation ◽  
Dimensional Wave ◽  
Do So

In this paper, we have tried to approach the concepts of two-dimensional wave equation and one dimensional heat equation through the means of the Navier Stoke’s equation for unsteady and incompressible flow. Our pursuit to do so has been supported with ample justifications and analytic discussions. The strong relation shared by the fluid dynamics, wave mechanics and heat flow has been brought to light through our attempts.

scholarly journals GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
GÜNTHER HÖRMANN ◽  
SANJA KONJIK ◽  
LJUBICA OPARNICA
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Beam Model ◽  
Order Partial Differential Equation ◽  
Viscoelastic Foundation ◽  
Regularization Techniques ◽  
Initial Boundary Value

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

2009 ◽  
Vol 20 (02) ◽  
pp. 313-322
Author(s):  
PILWON KIM
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Standard Method ◽  
Interpolation Method ◽  
Nonlinear Heat Equation ◽  
Geometric Integration ◽  
Numerical Schemes ◽  
New Class ◽  
Geometric Integrators

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

Free Vibration Analysis of Symmetrically Laminated Fully Clamped Skew Plates Using Extended Kantorovich Method

2011 ◽  
Vol 471-472 ◽  
pp. 739-744 ◽  
Author(s):  
Ali Fallah ◽  
Mohammad Hossein Kargarnovin ◽  
Mohammad Mohammadi Aghdam
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Closed Form Solution ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Power Series Solution ◽  
Order Partial Differential Equation ◽  
Kantorovich Method ◽  
Extended Kantorovich Method

In this paper, free vibration analysis of thin symmetrically laminated skew plates with fully clamped edges is investigated. The governing differential equation for skew plate which is a fourth order partial differential equation (PDE) is obtained by transforming the differential equation in Cartesian coordinates into skew coordinates. Based on the multi-term extended Kantorovich method (MTEKM) an efficient and accurate approximate closed-form solution is presented for the governing PDE. Application of the MTEKM reduces the governing PDE to a dual set of ordinary differential equations. These sets of equations are then solved with infinite power series solution, in an iterative manner until convergence was achieved. Results of this study show the fast rate of convergence of the MTEKM. Usually two or three iterations are enough to obtain reasonably accurate results. The frequency parameters of laminated composite plates are obtained for different skew angles and lay-up configuration for different composites laminates skew plates. Comparisons have been made with the available results in the literature which show the accuracy and efficiency of the method.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

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