A family of singular ordinary differential equations of third order with an integral boundary condition

We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem.


Introduction
We consider the following family of singular ordinary differential equations of third order with an integral boundary condition, indexed by a parameter λ ∈ R given by y (3) (t) − λ 2 y(t) = λ 2 √ π 1 t 3/2 , t > 0, y(0) = 1, y ′ (0) = 0, y (2) (1) = − λ √ π + λ 2 1 0 y(t)dt, where y (n) denotes the n−derivative of the function y. Singular boundary value problems arise very frequently in fluid mechanics and in other branches of applied mathematics. There are results on the existence and asymptotic estimates of solutions for third order ordinary differential equations with singularly perturbed boundary value problems, which depend on a small positive parameter see for example [16,19,27], on third order ordinary differential equations with singularly perturbed boundary value problems and with nonlinear coefficients or boundary conditions see for example [3,12,29,50], on third order ordinary differential equations with nonlinear boundary value problems see for example [18,28], on existence results for third order ordinary differential equations see for example [17,24], and particularly third order ordinary differential equations with integral boundary conditions see for example [2,6,7,20,21,39,42,47,49] In the last years there are several papers which consider integral or nonlocal boundary conditions on different branches of applications, e.g. for the heat equations see for example [10,13,14,15,22,26,30,34,35,36,38], for the wave equations [37], for the second order ordinary differential equations see for example [5,31,33,44,52,53,54], for the fourth order ordinary differential equations see for example [41,51], for higher order ordinary differential equations see for example [25], for fractional differential equations see for example [23,32,46].
Our goal is to prove in Section 2 that the system (1.1) is equivalent to the following Volterra integral equation of second kind which allows us to obtain the solution to some problems for nonclassical heat equation for any real parameter λ (see [4,8,9,11,40,43,45]).
In Section 3, we establish the dependence of the family of singular ordinary differential equations of third order (1.1) with respect to the parameter λ ∈ R by using the equivalence with the Volterra integral equation (1.2).

Equivalence and existence results
Preliminary, we give some results useful in the next sections.
Proof. The first three properties (2.1)-(2.3) follow from the the simple integration process.
To prove (2.4) we use the change of variable τ = σ + (t − σ)ξ then we obtain where B and Γ are the known Beta and Gamma functions defined by with the well known relations To prove (2.5) we use the same change of variable, so we obtain Proof. Firstly, we consider that y is a solution to the singular ordinary differential equation (1.1). Then, by using an integration in variable t we obtain And using the integral boundary condition so y (2) (0) = 0. Thus taking this new condition into account, from (2.6) by using an integration in variable t, the condition y ′ (0) = 0 and (2.1) we get Finally, from (2.7) by using a another integration in variable t, and the condition y(0) = 1, we obtain We can not arrive directly to the Volterra equation (1.2), but we can define the auxiliary function and now our goal is to prove that ϕ = 0. We have ϕ(0) = 0, by using the boundary y(0) = 1. Now, we compute the first derivative of ϕ using the property (2.3), we get From the other hand, by using (2.9), (2.7), (2.10) and the property (2.4) we obtain That is thus ϕ ′ (0) = 0. Therefore, we have and then we obtain thus ϕ (2) (0) = 0, and so on we obtain ϕ (n) (0) = 0 for all n ∈ N, then this part holds. Secondly, we consider that y is a solution of the Volterra integral equation (1.2), then we have the condition y(0) = 1 which is automatically satisfied.
for compactness expression.
Proof. By using the Adomian method [1,48] we propose, for the solution of the Volterra integral equation (1.2), the following serie of expansion functions given by and we obtain the following recurrence expansions : Then, following [9] we obtain (2.17) where I(t) and J(t) are given by (2.18) and (2.19) respectively, and the result holds.
The solution of the Volterra integral equation (1.2) is the key in order to obtain the solution of the following nonclassical heat conduction problem given by with a parameter λ ∈ R. Then the solution of the problem above is given by where U (t) is given by and g is the solution of the Volterra integral equation (1.2). Moreover, the heat flux on x = 0 is given by For the complet proof see [9].
3 Dependence of the solution with respect to λ From now on, we will consider that the solution to the singular ordinary differential equation of third order with an integral boundary condition (1.1) or equivalently the solution of the Volterra integral equation (1.2) depends also on the parameter λ ∈ R. We consider that t → g λ (t) be the solution of the Volterra integral equation (1.2) for the parameter λ. For ε ∈ (0, 1) be a fixed real number and T > 0, let consider the parameter λ such that and we define the norm g T = max 0≤t≤T |g(t)|.
Therefore, we obtain the following dependence results.
Proof. From the Volterra integral equation (1.2) we obtain and by using (3.1) follows (3.2). Moreover, consider g i (t) the solution of the Volterra integral equation (1.2) for λ i (i= 1, 2)), such that Then, we have Therefore, we get thus the result holds. Now, we obtain the dependence of the solution to the nonclassical heat conduction problem (2.20)-(2.22) with respect to the parameter λ. We consider that U λ and u λ are given respectively by Then, we obtain the following results: Theorem 3.2. We have the boundedness Moreover, the application λ → U λ (t), from [−λ ε,T , λ ε,T ] to C([0, T ]) is Lipschitzian. We have also the following boundedness the estimates

Conclusion
We have obtained the equivalence between a family of singular ordinary differential equations of third order with an integral boundary condition (1.1) and the Volterra integral equation (1.2) with a parameter λ ∈ R. We have also given the explicit solution of these equations and then some nonclassical heat conduction problems can be solved explicitely, for any real parameter λ. Finally, we have established the dependence of the family of singular differential equations of third order with respect to the parameter λ.