Chapter1-02
Chapter 1
Introduction
1.1 What is a differential equation? An equation relating a dependent variable to one or more independent variables by its derivative with respect to the independent variables is called a differential equation. Ordinary differential equation (ODE) has only one independent variable while partial differential equation (PDE) has two or more independent variables. Examples of ODE and PDE are:
= - 2(1.1-1)
+ 5+4u = 4excos(x)(1.1-2)
= c2( + ) (Two–dimensional heat equation)(1.1-3)
+ 3= 0(1.1-4)
A function u is called a solution to a partial differential equation whenever the equation becomes an identity in the independent variables upon substitution of u and its appropriate derivatives in the partial differential equation. For example if
= (1.1-5)
then u1(x,t) = sin(x)cos(t), u2(x,t) = sin(2x)cos(t),….., un(x,t) = sin(nx)cos(nt) are all solutions to Eq. (1-5).
One function u may possess many particular solutions, therefore auxiliary conditions are required to isolate or characterize an individual solution for a given problem. These individual solutions are usually composed from some subset of the totality of all particular solutions to a given equation.
Order of a differential equation is the order of the highest derivative that appears in the equation. Degree of a differential equation is the highest power of the highest order derivative that the equation contains.
Equation / Order / Degree(1.1-1) / Second / Second
(1.1-2) / Third / First
(1.1-3) / Second / First
(1.1-4) / First / First
A differential equation is called a non-linear equation if any products exist between the dependent variable and its derivative, or between the derivatives themselves, or between the dependent variables themselves.
Example:
= - 2non-linear(1.1-1)
+ 5+4u = 4excos(x)non-linear (1.1-2)
= c2( + ) linear(1.1-3)
+ 5sin(u) = 0non-linear
+ u = sin(x)linear
Many fields in engineering and the physical sciences require the study of ODE and PDE. Examples of those fields are acoustics, aerodynamics, elasticity, electrodynamics, fluid dynamics, geophysics (seismic wave propagation), heat transfer, meteorology, oceanography, optics, petroleum engineering, plasma physics (ionized liquids and gases), quantum mechanics.
1.2 One dimensional heat conduction equation. Conduction refers to energy transfer by molecular interactions. Energy carriers on the molecular level are 'electrons' and 'phonons' where the latter is a quantized lattice vibration. The interaction is a nearest-neighbor process that extends only a few molecular dimensions. Energy transport over a distance is by a staged transfer through molecular distances.
Fourier's law (1822), developed from observed phenomena, states that the rate of heat transfer in the 'n' direction is proportional to the temperature gradient
qn
where n is the direction of heat transfer and is the rate of change of distance in the direction n.
n is the unit normal vector, and t is the unit tangential vector with the following properties,
n= 1; t= 1; nt = 0; nn = 1; tt = 1
qn = C, where C = - Akn
qn = - knA
where
kn= thermal conductivity in 'n' direction, W/mK
A= area of surface perpendicular to n through which qn flows
The minus is a sign convention so that qn is positive in the direction it transfers. In this text, we will usually consider the isotropic materials where the thermal conductivity k is independent of direction. For one dimensional heat transfer in the x-direction only, the heat transfer rate is then
qx = - kA
or in terms of the heat flux
= - k
Consider a rod of constant cross sectional area A oriented in the x-direction (from x = 0 to x = L)
Let e(x, t) = thermal (heat) energy density = thermal energy per unit volume, (J/m3)
Q(x, t) = heat source = heat generated per unit volume per unit time, (W/m3)
Thermal energy in volume Ax = e(x, t) Ax, (J)
Conservation of energy
= A A + Q(x, t) Ax
Since A is a constant along x
= + Q(x, t)
= + Q(x, t)
e(x, t) = (x) Cp(x) T(x, t)
where(x)= mass density, and Cp(x) = specific heat
= k
The one-dimensional heat conduction equation becomes
= + Q(x, t)
For constant physical properties , Cp, and k
Cp = k + Q(x, t)
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