scholarly journals THE MATHEMATICAL MODELLING OF THE NONLINEAR HEAT TRANSPORT IN THIN PLATE

1999 ◽  
Vol 4 (1) ◽  
pp. 44-50
Author(s):  
A. Buikis
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Mathematical Modelling ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
First Order ◽  
Nonlinear Algebraic Equations

The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations.

scholarly journals Theory of differential offset continuation

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 718-732 ◽  
Author(s):  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Frequency Domain ◽  
Initial Value Problem ◽  
Data Transformation ◽  
Initial Value ◽  
Partial Differential ◽  
Reflection Data ◽  
Zero Offset

I introduce a partial differential equation to describe the process of prestack reflection data transformation in the offset, midpoint, and time coordinates. The equation is proved theoretically to provide correct kinematics and amplitudes on the transformed constant‐offset sections. Solving an initial‐value problem with the proposed equation leads to integral and frequency‐domain offset continuation operators, which reduce to the known forms of dip moveout operators in the case of continuation to zero offset.

A Finite Element Method for the Solution of Free Boundary Problem

2004 ◽  
Author(s):  
K. N. Rai ◽  
D. C. Rai
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Finite Element ◽  
Thermal Diffusivity ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Initial Value Problem ◽  
Algebraic Equations ◽  
Initial Value

A finite element method is presented for the solution of a free boundary problem which arises during planar melting of a semi-infinite medium initially at a temperature which is slightly below the melting temperature of the solid. The surface temperature is assumed to vary with time. Two different situations are considered (I) when thermal diffusivity is independent of temperature and (II) when thermal diffusivity varies linearly with temperature. The differential equation governing the process is converted to initial value problem of vector matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied, a linear differential equation can be represented by a set of linear algebraic equations and a nonlinear differential equation can be represented by a set of nonlinear algebraic equations. The solution of the problem is then found in terms of Chebyshev polynomial of second kind. The solution of this initial value problem is utilized iteratively in the interface heat flux equation to determine interface location as well as the temperature in two regions. The method appears to be accurate in cases for which closed form solutions are available, it agrees well with them. The effect of several parameters on the melting are analysed and discussed.

scholarly journals APPROXIMATE SOLUTION OF A NONLINEAR INTEGRO-DIFFERENTIAL EQUATION BY THE NEWTON-KANTOROVICH METHOD

2020 ◽  
pp. 609-614
Author(s):  
I.A. Usenov ◽  
Yu.V. Kostyreva ◽  
S. Almambet kyzy
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Approximate Solution ◽  
Initial Value Problem ◽  
Nonlinear Integral Equation ◽  
Initial Problem ◽  
Initial Value ◽  
Kantorovich Method ◽  
Integro Differential Equation ◽  
First Order

In this paper, we propose a method for studying the initial value problem for a first-order nonlinear integro-differential equation. The initial problem is reduced by substitution to a nonlinear integral equation with the Urson operator. To construct a solution to a nonlinear integral equation, the Newton-Kantorovich method is used.

SIMPLE ALGORITHM'S FOR THE CALCULATION OF HEAT TRANSPORT PROBLEM IN PLATE

2001 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
H. Kalis ◽  
I. Kangro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Transport Problem ◽  
Two Dimensional ◽  
Initial Value ◽  
Equation Solution ◽  
Averaging Methods ◽  
First Order

The approximations of some heat transport problem in a thin plate are based on the finite volume and conservative averaging methods [1,2]. These procedures allow one to reduce the two dimensional heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary differential equations (ODEs) of the first order or to an initial‐value problem for one ordinary differential equations of the first order with one algebraic equation. Solution of the corresponding problems is obtained by using MAPLE routines “gear”, “mgear” and “lsode”.

Estimates for Solutions of Wave Equations with Vanishing Curvature

1985 ◽  
Vol 37 (6) ◽  
pp. 1176-1200 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Linear Combination ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

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