In this Section, we will study some initial and boundary value problems for the non-classical heat equation in the domain

where *s* = *s*(*t*) is a continuous function of *t* over the interval t > 0 and *s*(0) = 1. The IBVP are reduced to equivalent systems of integral equations in order to get the existence of a solution.

We consider the following problem (P6):

The function *F* is now related to the evolution of the temperature instead of the heat flux at *x* = 0. The problem (P6) can be considered a non-classical moving boundary problem for the heat equation as a generalization of the moving boundary problem for the classical heat equation [13] which can be useful in the study of free boundary problems for the heat-diffusion equation [12].

We will use the Neumann function, which is defined by

### Theorem 14

Under the assumptions (HA) to (HD) the solution *u* to the problem (P6) has the expression

where the function *V*, defined by

and the piecewise continuous functions *ϕ*_{1} and *ϕ*_{2} must satisfy the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}are solutions to the integral system (7.9)-(7.11), and *u* has the expression (7.7), then *u* is a solution to the problem (P6). Moreover, *V*(*t*) = *u*(0*,t*) and the solution *u* is unique among the class of solutions for which *u*_{
x
}is bounded.

### Proof

We first make a smooth extension of *h* outside of 0≤*x*≤1, so that the extended h is bounded and has compact support. The solution *u* is now assumed to have the form (7.7), where *V*, *ϕ*_{1} and *ϕ*_{
2
}are unknown continuous functions that they are to be determined. Note that the initial condition (7.5) is satisfied. From the differential equation we obtain

and therefore by (7.8) the differential equation is satisfied. The system of integral equations is derived from the boundary conditions. The second equation is obtained allowing *x* to tend to *s*(*t*) and using the Lemma 14.2.3 of [13, page 218], i.e.,

Letting *x* to tend to zero in (7.7), we obtain the third equation, i.e.,

Now let us derive *u* with respect to *x* from (7.6) and we get,

When *x* tends to zero in (7.15), and using the jump formulae of the fundamental solution to the heat equation [15], we obtain

and the first integral equation holds. Consequently, if *u* possesses the form (6.7), then the functions *V*, *ϕ*_{1} and *ϕ*_{
2
}must satisfy the system (7.9) to (7.11).

Moreover, if the continuous functions *V*, *ϕ*_{1} and *ϕ*_{
2
}verify the system (7.9) to (7.11) for all 0 ≤ *t* ≤ *T*, then we can consider the expression (7.7) for *u*, which satisfies the initial condition (7.5). Allowing *x* to tend to zero in (7.15), and using (7.10) we obtain (7.8), and therefore the differential equation is satisfied. From Lemma 4.2.3 of [13, page 50] we see that

Hence, from (7.8) we have *u*_{
x
}(0*,t*) = *f*(*t*). Likewise, *u* assumes the value *g* as *x* tends to *s*(*t*), and therefore the equivalence between (7.3) to (7.6) and (7.9) to (7.11) holds.

Finally, in order to prove the uniqueness and existence of solution to the system of integral equations (7.9) to (7.11), we will verify hypothesis (8.2.40) to (8.2.44) of the Corollary 8.2.1 of [13, p. 91]. First we define the following functions:

Now we will prove (8.2.40) [13]. We have for *i* = 1,2,3:

For the first function we have,

and by using the classical inequality

we deduce that

where . For the second function, by using (HC3), we have

By using inequality (7.23), we can get

For the third function, by using (HC3), we have

and by using inequality (7.23), we get

If we define

the hypothesis (8.2.40) [13] is satisfied. Now let us prove (8.2.41) to (8.2.42) [13]. We have

where *C*_{4} and *C*_{5} are positive constants. Therefore we define the function *α* as follows:

which is an increasing function and tends to zero, when *η* tends to zero. Let us note that *H*_{
i
}(t,*τ*,0,0,0) = 0 for all *i* = 1, 2,3, and therefore hypothesis (8.2.43) and (8.2.44) [13] are satisfied.▀

Now, we can consider the following problem (P7):

In this case, the function *F* depends on the evolution of the temperature of the temperature *u*(0*,t*) on the fixed face *x* = 0 while a heat flux condition is given by (7.33). This non-classical problem (P7) can be consider as a complementary problem to the previous problem (P1) given by (1.1) to (1.4) in which the source term *F* depends on the heat flux on the fixed face *x* = 0 while a temperature boundary condition (1.2) is given on the face *x* = 0.

### Corollary 15

Under the same assumptions of Theorem 9, the solution *u* to the problem (P7) is given by the expression

and then the unknown function *V*, defined by (7.8), and the unknown piecewise continuous functions *ϕ*_{1} and *ϕ*_{
2
}are the solution to the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}are solutions to the integral system (7.37) to (7.39), and we define *u* by the expression (7.36), then *u* is a solution to the problem (P7). Moreover, we have *V*(*t*) = *u*(0*,t*).

### Theorem 16

Under the assumptions (HA) to (HD) the solution *u* to the problem (P8):

is given by:

where the unknown function *V*, defined by (7.8), and the unknown piecewise continuous functions *ϕ*_{1} and *ϕ*_{
2
}are solutions to the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}are solutions to the integral system (7.45) to (7.47), and *u* has the form (7.44), then *u* is a solution to the problem (P8). Moreover, we have *V*(*t*) = *u*(0*,t*).

### Proof

It is similar to the one given for Theorem 14.▀