Solution of the DCSF PDE

This appendix addresses the issue of solving the PDE (2.5).

A.1 Homogeneous Equation

image409 Подпись: (A.1)

Start by considering the homogeneous equation

to which the following ODE relating the independent variables t and x is associated


The first integral of (A.2) is a relation of the form

p(x, t) = C

Подпись: p(x, t) = x image413 Подпись: (A.3)

for C an arbitrary constant, satisfied by any solution x = x (t) of (A.2), where the function p is not identically constant for all the values of x and t. In other words, the function p is constant along each solution of (A.2), with the constant C depending on the solution. Since in the case of equation (a1e1) there are 2 independent variables, there is only one functionally independent integrals Ibragimov (1999). By integrating (A.2) with respect to time, its first integral is found to be

A function p(x, t) is a first integral of (A.2) if and only if it is a solution of the homogeneous PDE (A.1). Furthermore, or t > t0, the general solution of (A.1) is given by

T(x, t) = F(P(x, t)), (A.4)

where F is an arbitrary function Ibragimov (1999). The general solution of (A.1) is thus given by


T(x, t) = F(x – u(a)da), (A.5)


where F(x) for x є [0, L] is the temperature distribution along the pipe at t = t0.

A.2 Non-homogeneous Equation

Consider now the non-homogeneous Eq. (2.5) that, for the sake of simplifying the notation, is written as

9 9

T(z, t) = u T(z, t) + g(x, t, T), (A.6)

91 9 z


g(x, t, T) := aR(t) – у T(z, t). (A.7)

To solve (A.6), and according to a procedure known as Laplace’s method Ibragimov (1999), introduce the new set of coordinates (^, t) given by

Подпись:Подпись:f = ni(x, t) := p(x, t)


t = n2(x, t) := t

where p is the first integral (A.3). The chain rule for derivatives yields

9 T

_ dpi

9 T dp2

Ji + ~9x ‘

9 T

9 x

9 x


9 T

_ dpi

9 T 9p2

‘ + ~df

9 T


= a t


Since the transformation of variables is defined by (A.8, A.9), these equations reduce to

Подпись: (A.12) (A.13) dT _ dp 9T 9x 9x df ’

9T _ dp dT dT

at at af dr

Подпись: dp dp — + u — dt dx image419 Подпись: (A.14)

Therefore, in the new coordinates, and regrouping terms, the non-homogeneous equa­tion (a1e7) is written

Since p satisfies the homogeneous equation (A.1) and r = t, this equation reduces to the ODE


= g (A.15)


where x is to be expressed in terms of t and f by solving (A.8) with respect to x. Since g is given by (A.7), the above ODE reads


— Tf t) = -YTf t) + aR(t). (A.16)

Equation(A.16) is a linear, scalar, first order ODE whose solution is


T(f’ t) = T(f’ t0)e-y(t-t0) + a j R(a)ey(a-t)da. (A.17)


Подпись: □Inverting the change of variable (A.8) and using (A.3) yields (2.6).

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