A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations

Abstract

The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved.

1. Introduction

Integro-differential equations arise in many areas in sciences and engineering. For example, Fredholm and Volterra integro-differential equations are used in [1,2,3] for modeling neural networks; Fredholm integro-differential equations are utilized in [4,5] to model the response of plates and shells in the theory of elasticity; Volterra integro-differential equations are employed in [6,7,8] to simulate thermal problems in the glass-forming process, drug concentrations in pharmacokinetics, and Hepatitis B Virus infection in medicine, respectively.

This paper is concerned with the factorization and exact solution of boundary value problems for a class of n-th order Fredholm integro-differential equations of the second type with nonlocal boundary conditions. Closed form solutions are apparently preferred in all cases [9], but they cannot be accomplished for many real-life problems due to their mathematical complexities. For the exact solution of linear Fredholm integro-differential equations the direct computation method can be used, but its application is limited to special cases where the kernel is separable, and the integrals involved are evaluated analytically [10]. Instead, approximate numerical methods are used. For the solution of integro-differential equations various approximate methods have been developed, see [11,12,13,14,15,16,17] and many others. Higher order integro-differential equations are often encountered in modeling and there have been the subject of numerous investigations [18,19,20,21]. When model accuracy is very important then boundary conditions must be nonlocal, i.e., involving combinations of various points and integrals. For instance, the necessity of integral conditions in certain models of epidemics and population growth and the effects when simplifying them are explained in [22]. Nonlocal boundary value problems for integro-differential equations have received much attention recently, see [23,24,25,26,27,28] and the references therein. Factorization methods are very important in constructing exact solutions, but their applicability is confined to certain kinds of operators and moreover they cannot be universal for all problems. They have been studied extensively for decomposing differential operators [29,30,31,32,33] and very little in factoring integro-differential operators [34,35,36,37]. Here, we continue the work in [35,36,37] and present a method for factorizing and solving yet another type of boundary value problems.

Specifically, in [35], in a Hilbert space H, the authors studied the factorability of the linear operator $B:H\to H$ of the type

$$Bu=A_2{A}_{1}u-S{〈A_{1}u,\mathcal{F}〉}_{H^m}-G{〈A_2{A}_{1}u,F〉}_{H^m}$$

into $B=B_2{B}_1$, where $B_1,B_2:H\to H$ are two lower-order linear operators, and the unique solvability of the boundary value problem $Bu=f$. The linear operators $A_1,A_2:H\to H$ are correct, i.e., $A_1,A_2$ are bijective and the associated inverse operators $A_1^{-1},A_2^{-1}$ are bounded on H, the operator $A_2$ is densely defined on H, the vectors $S=(s_1,\dots ,s_m),G=(g_1,\dots ,g_m),\mathcal{F}=col({\mathcal{F}}_1,\dots ,{\mathcal{F}}_m),F=col(F_1,\dots ,F_m)\in H^m$ and the elements ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_m$ are linearly independent, $f\in H$, and $u\in D\left(B\right)$ is unknown. In [36], the exact solution of the boundary value problem

$$Bu=B_1^{2}u=f,$$

in a Banach space X was investigated. The linear operator $B:X\to X$ is a self-composition of the operator $B_1:X\to X$,

$$B_{1}u=Au-G\mathsf{\Psi }\left(Au\right),D\left(B_1\right)=\{u\in D\left(A\right):\mathsf{\Theta }\left(u\right)=N\mathsf{{\rm Y}}\left(u\right)\},$$

where $A:X\to X$ is a maximal n-th order linear ordinary differential operator, $G=(g_1,\dots ,g_m)\in X^m$, $\mathsf{\Psi }=\mathrm{col}({\mathsf{\Psi }}_1,\dots ,{\mathsf{\Psi }}_m)$ is a vector of m linear bounded functionals on X, $\mathsf{\Theta }=\mathrm{col}({\mathsf{\Theta }}_1,\dots ,{\mathsf{\Theta }}_n),\mathsf{{\rm Y}}=\mathrm{col}({\mathsf{{\rm Y}}}_1,\dots ,{\mathsf{{\rm Y}}}_m)$ are vectors of bounded linear boundary functionals, N is a $n\times m$ constant matrix, $f\in X$, and $u\in D\left(B\right)$ is unknown. The paper [37] deals with the decomposition and the exact solution of the boundary value problem

$$Bu=Au-S\mathsf{\Phi }\left(u\right)-G\mathsf{\Psi }\left(A_{1}u\right)=f,D\left(B\right)=D\left(A\right),$$

where $B:X\to X$ is a linear operator, $A,A_1:X\to X$ are linear differential operators of order n and $n_1$, respectively, $S=(s_1,\dots ,s_m),G=(g_1,\dots ,g_m)\in X^m$, $\mathsf{\Phi }=\mathrm{col}({\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_m),\mathsf{\Psi }=\mathrm{col}({\mathsf{\Psi }}_1,\dots ,{\mathsf{\Psi }}_m)$ are vectors of linear bounded functionals on X, $f\in X$, and $u\in D\left(B\right)$.

The strict assumptions that were made and the requirement that the operators $A_1,A_2$ must be known in advance in [35] are restrictive in implementing the factorization technique in solving practical boundary value problems. On the other hand, the formulation in Banach space in [36,37] for solving the corresponding two problems above has been proved more convenient. Here, we generalize the results in [35] in Banach space and at the same time loosen the requirements. Further, we provide a step by step procedure for the factorization and solution of the boundary value problem.

In particular, in a Banach space X of complex valued functions of x defined on the region $\mathsf{\Omega }\subseteq {\mathbb{R}}^k,k\in \mathbb{N},k\ge 1$, we consider the following boundary value problem

$$\begin{array}{ccc}Bu& =& Au-\sum _{j=1}^m{\int }_{\mathsf{\Omega }}{K}_{1j}(x,t)A_{1}u\left(t\right)dt-\sum _{j=1}^m{\int }_{\mathsf{\Omega }}{K}_j(x,t)Au\left(t\right)dt=f,x\in \mathsf{\Omega },\hfill \\ D\left(B\right)& =& D\left(A\right)\subset X,\hfill \end{array}$$

(1)

where $B:X\to X$ is a linear operator with its domain $D\left(B\right)$ specified by the prescribed multipoint and integral conditions, $A:X\to X$ is a bijective linear differential operator of order n defined on the same domain as the operator B, $A_1:X\to X$ is a bijective linear differential operator of order $n_1<n$ whose domain $D\left(A_1\right)$ is determined by some multipoint and integral conditions, $K_{1j}(x,t),K_j(x,t)\in X(\mathsf{\Omega }\times \mathsf{\Omega })$ are known kernel functions, $f\left(x\right)\in X$ is an input function and $u\left(x\right)\in D\left(B\right)$ is the unknown function. We first examine under which conditions this problem can be decomposed into two simpler boundary value problems and then by solving these lower-order problems we derive its unique solution in closed form.

In addition, we investigate the factorization and solution of the boundary value problem

$$\begin{array}{ccc}Bu& =& A^{2}u-\sum _{j=1}^m{\int }_{\mathsf{\Omega }}{K}_{1j}(x,t)Au\left(t\right)dt-\sum _{j=1}^m{\int }_{\mathsf{\Omega }}{K}_j(x,t)A^{2}u\left(t\right)dt=f,x\in \mathsf{\Omega },\hfill \\ D\left(B\right)& =& D\left(A^2\right)\subset X,\hfill \end{array}$$

(2)

which is a special case of the boundary value problem (1).

The paper is organized as follows. Some preliminaries are given in Section 2. Next, the main results are presented in Section 3. In Section 4, two examples problems, one for ordinary and one for partial integro-differential equations, are solved to elucidate the implementation and highlight the efficiency of the procedure. Finally, in Section 5 some conclusions are given.

2. Materials and Methods

For the rest of the paper, we assume that the kernels are separable, i.e.,

$$K_{1j}(x,t)=s_j\left(x\right)h_{1j}\left(t\right),K_j(x,t)=g_j\left(x\right)h_j\left(t\right),j=1,\dots ,m,$$

where $s_j\left(x\right),g_j\left(x\right),h_{1j}\left(t\right),h_j\left(t\right)\in X$, and write the integro-differential Equation (1) in the symbolic form

$$Bu=Au-S\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(Au\right),D\left(B\right)=D\left(A\right),$$

(3)

where the vectors of functions

$$S=\left(\begin{array}{ccc}{s}_1& \cdots & s_m\end{array}\right),G=\left(\begin{array}{ccc}{g}_1& \cdots & g_m\end{array}\right),$$

and the vectors of functionals, for $w\left(x\right)\in X$,

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left(w\right)& =& \left(\begin{array}{c}{\mathsf{\Phi }}_1\left(w\right)\\ ⋮\\ {\mathsf{\Phi }}_m\left(w\right)\end{array}\right)=\left(\begin{array}{c}{\int }_{\mathsf{\Omega }}{h}_{11}\left(t\right)w\left(t\right)dt\\ ⋮\\ {\int }_{\mathsf{\Omega }}{h}_{1m}\left(t\right)w\left(t\right)dt\end{array}\right),\hfill \\ \hfill \mathsf{\Psi }\left(w\right)& =& \left(\begin{array}{c}{\mathsf{\Psi }}_1\left(w\right)\\ ⋮\\ {\mathsf{\Psi }}_m\left(w\right)\end{array}\right)=\left(\begin{array}{c}{\int }_{\mathsf{\Omega }}{h}_1\left(t\right)w\left(t\right)dt\\ ⋮\\ {\int }_{\mathsf{\Omega }}{h}_m\left(t\right)w\left(t\right)dt\end{array}\right).\hfill \end{array}$$

It is noted that $\mathsf{\Psi }\left(G\right)$ denotes the $m\times m$ matrix

$$\mathsf{\Psi }\left(G\right)=\left(\begin{array}{ccc}{\mathsf{\Psi }}_1\left(g_1\right)& \cdots & {\mathsf{\Psi }}_1\left(g_m\right)\\ ⋮& \ddots & ⋮\\ {\mathsf{\Psi }}_m\left(g_1\right)& \cdots & {\mathsf{\Psi }}_m\left(g_m\right)\end{array}\right),$$

where the element ${\mathsf{\Psi }}_i\left(g_j\right)$ is the value of the functional ${\mathsf{\Psi }}_i$ on the element $g_j$. It is easy to show that for an $m\times k$ constant matrix N,

$$\mathsf{\Psi }\left(GN\right)=\mathsf{\Psi }\left(G\right)N.$$

We recall that a linear operator $T:X\to X$ is injective if for every $u_1,u_2\in D\left(T\right)$, $u_1\ne u_2$ implies $T{u}_1\ne T{u}_2$. The operator T is surjective if $R\left(T\right)=X$. If T is both injective and surjective, then the operator is called bijective and there be the inverse operator $T^{-1}:X\to X$ defined by $T^{-1}f=u$ if and only if $Tu=f$ for each $f\in X$. The operator T is said to be correct if it is bijective and the inverse operator $T^{-1}$ is bounded on X. The problem $Tu=f$ is correct if the operator T is correct.

Finally, we will use several times the following Theorem, which has been shown in [21] and it is recalled here but with a different notation tailored to the needs of the present article.

Theorem 1.

Let X be a complex Banach space, $A:X\to X$ a bijective linear operator, $G=(g_1,\dots ,g_m)\in X^m$, $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\dots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$, and $T:X\to X$ the linear operator

$$Tu=Au-G\mathsf{\Psi }\left(Au\right),D\left(T\right)=D\left(A\right).$$

Then the following statements are true:

(i) 

The operator T is bijective on X if and only if

$$detW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0,$$

and the unique solution to boundary value problem $Tu=f$, for any $f\in X$, is given by the formula

$$u=T^{-1}f=A^{-1}f+A^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right).$$

(ii) 

If in addition the operator $A^{-1}$ is bounded on X, then T is correct.

3. Main Results

First, we show the following two Lemmas which we will need to prove the main Theorem below.

Lemma 1.

Let the linear integro-differential operator $B:X\to X$ be defined by

$$Bu=Au-S\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(Au\right),u\in D\left(B\right)=D\left(A\right),$$

where $A:X\to X$ is a linear differential operator of order n, $A_1:X\to X$ is a linear differential operator of order $n_1$ with $n_1<n$, $G=(g_1,\dots ,g_m)$, $S=(s_1,\dots ,s_m)\in X^m$, and $\mathsf{\Phi }=col({\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_m)$, $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\cdots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$.

If there exist a linear differential operator $A_2:X\to X$ of order $n_2=n-n_1$ such that

$$A=A_2{A}_1,D\left(B\right)=D\left(A\right)=D\left(A_2{A}_1\right),$$

(4)

and a vector $P=(p_1,\cdots ,p_m)\in {\left[D\left(A_2\right)\right]}^m$ satisfying the equation

$$A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right)=S,$$

(5)

then the operator B can be decomposed into

$$Bu=B_2{B}_{1}u,$$

where the two linear integro-differential operators $B_1,B_2:X\to X$ are defined by

$$\begin{array}{ccc}\hfill B_{1}u& =& A_{1}u-P\mathsf{\Phi }\left(A_{1}u\right),D\left(B_1\right)=D\left(A_1\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill B_{2}u& =& A_{2}u-G\mathsf{\Psi }\left(A_{2}u\right),D\left(B_2\right)=D\left(A_2\right).\hfill \end{array}$$

(7)

Proof. 

Suppose there exist an $n_2$-order differential operator $A_2$ and a vector P satisfying (4) and (5).

Then by using the definition of B, we can construct the operators $B_1$ and $B_2$ in (6) and (7), respectively.

Moreover, we have

$$\begin{array}{ccc}\hfill Bu& =& A_2{A}_{1}u-S\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(A_2{A}_{1}u\right)\hfill \\ & =& A_2{A}_{1}u-(A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right))\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(A_2{A}_{1}u\right)\hfill \\ & =& (A_2{A}_{1}u-G\mathsf{\Psi }\left(A_2{A}_{1}u\right))-(A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right))\mathsf{\Phi }\left(A_{1}u\right)\hfill \\ & =& B_2\left(A_{1}u\right)-B_2\left(P\mathsf{\Phi }\left(A_{1}u\right)\right)\hfill \\ & =& B_2(A_{1}u-P\mathsf{\Phi }\left(A_{1}u\right))\hfill \\ & =& B_2{B}_{1}u,u\in D\left(B\right).\hfill \end{array}$$

Further,

$$\begin{array}{ccc}\hfill D\left(B_2{B}_1\right)& =& \{u\in D\left(B_1\right):B_{1}u\in D\left(B_2\right)\}\hfill \\ & =& \{u\in D\left(A_1\right):A_{1}u-P\mathsf{\Phi }\left(A_{1}u\right)\in D\left(A_2\right)\}\hfill \\ & =& \{u\in D\left(A_1\right):A_{1}u\in D\left(A_2\right)\}\hfill \\ & =& D\left(A_2{A}_1\right)=D\left(A\right)=D\left(B\right),\hfill \end{array}$$

which completes the proof. □

Lemma 2.

Let the $n_2$-order linear differential operator $A_2:X\to X$ be a bijective operator, $G=(g_1,\dots ,g_m)\in X^m$ and $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\cdots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$. Then the operator

$$B_{2}u=A_{2}u-G\mathsf{\Psi }\left(A_{2}u\right),D\left(B_2\right)=D\left(A_2\right),$$

is bijective on X if and only if

$$detW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0.$$

In this case, the unique solution of the integro-differential equation

$$B_{2}P=A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right)=S,foranyS=(s_1,\dots ,s_m)\in X^m,$$

where $P=(p_1,\cdots ,p_m)\in {\left[D\left(A_2\right)\right]}^m$, is given by

$$P=B_2^{-1}S=A_2^{-1}S+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(S\right).$$

Proof. 

It can be shown easily by applying Theorem 1. □

Theorem 2.

Let the linear integro-differential operator $B:X\to X$ be defined by

$$Bu=Au-S\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(Au\right),u\in D\left(B\right)=D\left(A\right),$$

(8)

where $A:X\to X$ is a linear differential operator of order n, $A_1:X\to X$ is a bijective linear differential operator of order $n_1$ with $n_1<n$, $G=(g_1,\dots ,g_m)$, $S=(s_1,\dots ,s_m)\in X^m$, and $\mathsf{\Phi }=col({\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_m)$, $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\cdots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$. Let there be a bijective linear differential operator $A_2:X\to X$ of order $n_2=n-n_1$ such that

$$A=A_2{A}_1,D\left(B\right)=D\left(A\right)=D\left(A_2{A}_1\right).$$

(9)

Then the following statements are true.

(i) 

If

$$detW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0,$$

(10)

then the operator B can be factorized into $Bu=B_2{B}_{1}u$ where the two linear integro-differential operators $B_1,B_2:X\to X$ are defined by

$$\begin{array}{ccc}\hfill B_{1}u& =& A_{1}u-P\mathsf{\Phi }\left(A_{1}u\right),D\left(B_1\right)=D\left(A_1\right),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill B_{2}u& =& A_{2}u-G\mathsf{\Psi }\left(A_{2}u\right),D\left(B_2\right)=D\left(A_2\right),\hfill \end{array}$$

(12)

where

$$P=A_2^{-1}S+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(S\right).$$

(13)

(ii) 

If there exists a vector $P=(p_1,\cdots ,p_m)\in {\left[D\left(A_2\right)\right]}^m$ satisfying the equation

$$A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right)=S,$$

(14)

then the operator B is bijective if and only if

$$detV=det[I_m-\mathsf{\Phi }\left(P\right)]\ne 0anddetW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0.$$

(15)

In this case, the unique solution to the boundary value problem

$$Bu=f,\mathrm{for\; any}f\in X,$$

(16)

is given by

$$\begin{array}{ccc}\hfill u& =& B_1^{-1}f\hfill \\ & =& A_1^{-1}\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right)\hfill \\ & & +A_1^{-1}P{V}^{-1}\mathsf{\Phi }\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right).\hfill \end{array}$$

(17)

Proof. 

(i) If Equation (10) holds, then from Lemma 2 it is implied that the operator $B_2$ defined in (12) is bijective and that the unique solution of the boundary value problem $B_{2}P=A_{2}P-G\mathsf{\Psi }\left(A_{2}P\right)=S$ is given in (13). Then, from Lemma 1 it follows that the operator B can be decomposed into $Bu=B_2{B}_{1}u$ where the operators $B_1$ and $B_2$ are given in (11) and (12), respectively.

(ii) From Lemma (1) it is deduced that $Bu=B_2{B}_{1}u$ with $B_1$ and $B_2$ given by (11) and (12), respectively, and $B_{2}P=S$. Substituting into (8), we have

$$\begin{array}{ccc}\hfill Bu& =& Au-S\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(Au\right)\hfill \\ & =& Au-B_{2}P\mathsf{\Phi }\left(A_{1}u\right)-G\mathsf{\Psi }\left(Au\right)\hfill \\ & =& Au-\left(\begin{array}{cc}{B}_{2}P& G\end{array}\right)\left(\begin{array}{c}\mathsf{\Phi }\left(A_{1}u\right)\\ \mathsf{\Psi }\left(Au\right)\end{array}\right)\hfill \\ & =& Au-\left(\begin{array}{cc}{B}_{2}P& G\end{array}\right)\left(\begin{array}{c}\mathsf{\Phi }\left(A_2^{-1}{A}_2{A}_{1}u\right)\\ \mathsf{\Psi }\left(Au\right)\end{array}\right)\hfill \\ & =& Au-\left(\begin{array}{cc}{B}_{2}P& G\end{array}\right)\left(\begin{array}{c}\mathsf{\Phi }\left(A_2^{-1}Au\right)\\ \mathsf{\Psi }\left(Au\right)\end{array}\right)\hfill \\ & =& Au-\widehat{G}\widehat{\mathsf{\Psi }}\left(Au\right),\hfill \end{array}$$

where

$$\widehat{G}=\left(\begin{array}{cc}{B}_{2}P& G\end{array}\right),\widehat{\mathsf{\Psi }}\left(Au\right)=\left(\begin{array}{c}\mathsf{\Phi }\left(A_2^{-1}Au\right)\\ \mathsf{\Psi }\left(Au\right)\end{array}\right).$$

By applying Theorem 1, the operator B is bijective if and only if

$$\begin{array}{ccc}\hfill det\widehat{W}& =& det\left[I_{2m}-\widehat{\mathsf{\Psi }}\left(\widehat{G}\right)\right]\hfill \\ & =& det\left[\left[\begin{array}{cc}{I}_m& 0_m\\ 0_m& I_m\end{array}\right]-\left[\begin{array}{cc}\mathsf{\Phi }\left(A_2^{-1}{B}_{2}P\right)& \mathsf{\Phi }\left(A_2^{-1}G\right)\\ \mathsf{\Psi }\left(B_{2}P\right)& \mathsf{\Psi }\left(G\right)\end{array}\right]\right]\hfill \\ & =& det\left[\begin{array}{cc}{I}_m-\mathsf{\Phi }\left(A_2^{-1}{B}_{2}P\right)& -\mathsf{\Phi }\left(A_2^{-1}G\right)\\ -\mathsf{\Psi }\left(B_{2}P\right)& I_m-\mathsf{\Psi }\left(G\right)\end{array}\right]\hfill \\ & =& det\left[\begin{array}{cc}{I}_m-\mathsf{\Phi }\left(P\right)+\mathsf{\Phi }\left(A_2^{-1}G\right)\mathsf{\Psi }\left(A_{2}P\right)& -\mathsf{\Phi }\left(A_2^{-1}G\right)\\ -\mathsf{\Psi }\left(A_{2}P\right)+\mathsf{\Psi }\left(G\right)\mathsf{\Psi }\left(A_{2}P\right)& I_m-\mathsf{\Psi }\left(G\right)\end{array}\right]\ne 0.\hfill \end{array}$$

Multiplying from the right the elements of the second column by $\mathsf{\Psi }\left(A_{2}P\right)$ and adding to the corresponding elements of the first column, we obtain

$$\begin{array}{ccc}\hfill det\widehat{W}& =& det\left[\begin{array}{cc}{I}_m-\mathsf{\Phi }\left(P\right)& -\mathsf{\Phi }\left(A_2^{-1}G\right)\\ 0_m& I_m-\mathsf{\Psi }\left(G\right)\end{array}\right]\hfill \\ & =& detVdetW\ne 0.\hfill \end{array}$$

So we proved that the operator B is bijective if and only if the two conditions in (15) are fulfilled.

By means of Lemma 1 the boundary value problem (16) degenerates to $B_2{B}_{1}u=f$ where the operators $B_1$ and $B_2$ are given in (11) and (12), respectively. We set $B_{1}u=v$ and factor the problem to two simpler boundary value problems

$$\begin{array}{ccc}\hfill B_{1}u& =& v,D\left(B_1\right)=D\left(A_1\right),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill B_{2}v& =& f,D\left(B_2\right)=D\left(A_2\right).\hfill \end{array}$$

(19)

From Lemma 2 it follows that both operators $B_1$ and $B_2$ are bijective. Thus, we first solve the boundary value problem in (19) by using Theorem 1 to obtain

$$v=B_2^{-1}f=A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right),$$

and then substitute into (18) and solve again by Theorem 1 to obtain

$$\begin{array}{ccc}\hfill u& =& B_1^{-1}\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right)\hfill \\ & =& A_1^{-1}\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right)\hfill \\ & & +A_1^{-1}P{V}^{-1}\mathsf{\Phi }\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right).\hfill \end{array}$$

This completes the proof. □

The boundary value problem in (2) can be factorized and solved in a similar manner. For this, we state the following Lemma and Theorem, which are analogous to Lemma 1 and Theorem 2, respectively, and can be proved in a similar way.

Lemma 3.

Let the linear integro-differential operator $B:X\to X$ be defined by

$$Bu=A^{2}u-S\mathsf{\Phi }\left(Au\right)-G\mathsf{\Psi }\left(A^{2}u\right),u\in D\left(B\right)=D\left(A^2\right),$$

where $A:X\to X$ is a linear differential operator of order n, $G=(g_1,\dots ,g_m)$, $S=(s_1,\dots ,s_m)\in X^m$, and $\mathsf{\Phi }=col({\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_m)$, $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\cdots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$. If there exists a vector $P=(p_1,\cdots ,p_m)\in {\left[D\left(A\right)\right]}^m$ satisfying the equation

$$AP-G\mathsf{\Psi }\left(AP\right)=S,$$

then the operator B can be decomposed into

$$Bu=B_2{B}_{1}u,$$

where the two linear integro-differential operators $B_1,B_2:X\to X$ are defined by

$$\begin{array}{ccc}\hfill B_{1}u& =& Au-P\mathsf{\Phi }\left(Au\right),D\left(B_1\right)=D\left(A\right),\hfill \\ \hfill B_{2}u& =& Au-G\mathsf{\Psi }\left(Au\right),D\left(B_2\right)=D\left(A\right).\hfill \end{array}$$

Theorem 3.

Let the linear integro-differential operator $B:X\to X$ be defined by

$$Bu=A^{2}u-S\mathsf{\Phi }\left(Au\right)-G\mathsf{\Psi }\left(A^{2}u\right),u\in D\left(B\right)=D\left(A^2\right),$$

where $A:X\to X$ is a bijective linear differential operator of order n, $G=(g_1,\dots ,g_m)$, $S=(s_1,\dots ,s_m)\in X^m$, and $\mathsf{\Phi }=col({\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_m)$, $\mathsf{\Psi }=col({\mathsf{\Psi }}_1,\cdots ,{\mathsf{\Psi }}_m)\in {\left[X^{*}\right]}^m$. Then the following statements are true.

(i) 

If

$$detW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0,$$

then the operator B can be factorized into $Bu=B_2{B}_{1}u$ where the two linear integro-differential operators $B_1,B_2:X\to X$ are defined by

$$\begin{array}{ccc}\hfill B_{1}u& =& Au-P\mathsf{\Phi }\left(Au\right),D\left(B_1\right)=D\left(A\right),\hfill \\ \hfill B_{2}u& =& Au-G\mathsf{\Psi }\left(Au\right),D\left(B_2\right)=D\left(A\right),\hfill \end{array}$$

and

$$P=A^{-1}S+A^{-1}G{W}^{-1}\mathsf{\Psi }\left(S\right).$$

(ii) 

If there exists a vector $P=(p_1,\cdots ,p_m)\in {\left[D\left(A\right)\right]}^m$ satisfying the equation

$$AP-G\mathsf{\Psi }\left(AP\right)=S,$$

then the operator B is bijective if and only if

$$detV=det[I_m-\mathsf{\Phi }\left(P\right)]\ne 0anddetW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0.$$

In this case, the unique solution to the boundary value problem

$$Bu=f,\mathrm{for} \mathrm{any}f\in X,$$

is given by

$$\begin{array}{ccc}\hfill u& =& B^{-1}f\hfill \\ & =& A^{-1}\left(A^{-1}f+A^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right)\hfill \\ & & +A^{-1}P{V}^{-1}\mathsf{\Phi }\left(A^{-1}f+A^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right).\hfill \end{array}$$

Finally, in Listing 1, we present the steps to be followed in factoring and solving the boundary value problem (16).

Listing 1. Steps for factoring and solving the boundary value problem Bu = f in (16).

1. Specify the operator B corresponding to the given integro-differential equation and boundary conditions. 2. Define the operators $A,A_1$ of order n and $n_1$, respectively. 3. Determine the operator $A_2$ of order $n_2=n-n_1$ by requiring the fulfillment of the condition in (9), viz. $$A=A_2{A}_1,D\left(B\right)=D\left(A\right)=D\left(A_2{A}_1\right).$$ 4. Check the bijectivity of $A_1$ and $A_2$ and find the corresponding inverse operators $A_1^{-1}$ and $A_2^{-1}$. 5. Specify m and set up the vectors of functions $S,G$ and the vectors of functionals $\mathsf{\Phi },\mathsf{\Psi }$. 6. Compute W and ensure the satisfaction of the second condition in (15), viz. $$detW=det[I_m-\mathsf{\Psi }\left(G\right)]\ne 0.$$ 7. Compute P by using (13), i.e., $$P=A_2^{-1}S+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(S\right).$$ 8. Compute V and check upon the fulfillment of the first condition in (15), viz. $$detV=det[I_m-\mathsf{\Phi }\left(P\right)]\ne 0.$$ 9. Compute the unique solution to the given problem by substituting into (17), namely $$u=A_1^{-1}\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right)+A_1^{-1}P{V}^{-1}\mathsf{\Phi }\left(A_2^{-1}f+A_2^{-1}G{W}^{-1}\mathsf{\Psi }\left(f\right)\right).$$

4. Examples

To show the application and the effectiveness of the method described in the previous Section, we solve two illustrative example boundary value problems. First, an ordinary integro-differential equation with nonlocal boundary conditions is considered. Next, an integro-partial differential equation with integral conditions is factorized and solved in closed form.

In the Appendix A, we present the implementation of the procedure in the open source software Maxima.

Example 1.

Consider the ordinary integro-differential equation

$$u^{‴}\left(x\right)-8x^2{\int }_0^{1}t{u}^{\prime }\left(t\right)dt-(3x+1){\int }_0^1{t}^2{u}^{‴}\left(t\right)dt=f\left(x\right),0<x<1,$$

(20)

subject to nonlocal boundary conditions

$$u\left(0\right)=2{\int }_0^{1}u\left(t\right)dt,u^{\prime }\left(0\right)=-u^{\prime }\left(1\right),u^{″}\left(0\right)=-u^{″}\left(1\right),$$

(21)

for any $f\left(x\right)\in C[0,1]$.

1. 

According to procedure in Listing 1, we take $X=C[0,1]$ and define the operator $B:C[0,1]\to C[0,1]$ by

$$\begin{array}{ccc}Bu& =& u^{‴}\left(x\right)-8x^2{\int }_0^{1}t{u}^{\prime }\left(t\right)dt-(3x+1){\int }_0^1{t}^2{u}^{‴}\left(t\right)dt,\hfill \\ D\left(B\right)& =& \left\{u\in C^3[0,1]:u\left(0\right)=2{\int }_0^{1}u\left(t\right)dt,u^{\prime }\left(0\right)=-u^{\prime }\left(1\right),u^{″}\left(0\right)=-u^{″}\left(1\right)\right\}.\hfill \end{array}$$

2. 

Further, we define the operators $A,A_1:C[0,1]\to C[0,1]$ of order $n=3$ and $n_1=1$, respectively, as

$$\begin{array}{ccc}\hfill Au& =& u^{‴}\left(x\right),D\left(A\right)=D\left(B\right),\hfill \\ \hfill A_{1}u& =& u^{\prime }\left(x\right),D\left(A_1\right)=\left\{u\in C^1[0,1]:u\left(0\right)=2{\int }_0^{1}u\left(t\right)dt\right\}.\hfill \end{array}$$

3. 

The operator $A_2:C[0,1]\to C[0,1]$ is determined by requiring

$$\begin{array}{ccc}Au& =& u^{‴}\left(x\right)=A_2{A}_{1}u=A_2\left(u^{\prime }\left(x\right)\right)=A_{2}v=v^{″}\left(x\right),v=u^{\prime }\left(x\right),\hfill \\ D\left(A_2{A}_1\right)& =& \left\{u\in D\left(A_1\right):A_{1}u=u^{\prime }\left(x\right)=v\in D\left(A_2\right)\right\}\hfill \\ & =& \left\{u\in C^1[0,1]:u\left(0\right)=2{\int }_0^{1}u\left(t\right)dt,u^{\prime }\left(x\right)=v\in D\left(A_2\right)\right\}\hfill \\ & =& D\left(B\right),\hfill \end{array}$$

(22)

which implies that

$$A_{2}v=v^{″}\left(x\right),D\left(A_2\right)=\left\{v\in C^2[0,1]:v\left(0\right)=-v\left(1\right),v^{\prime }\left(0\right)=-v^{\prime }\left(1\right)\right\}.$$

4. 

The operators $A_1$ and $A_2$ are known to be correct and their inverses for any $f\left(x\right)\in C[0,1]$ are given by

$$\begin{array}{ccc}\hfill A_1^{-1}f& =& 2{\int }_0^1(t-1)f\left(t\right)dt+{\int }_0^{x}f\left(t\right)dt,\hfill \\ \hfill A_2^{-1}f& =& \frac{1}{2}{\int }_0^1\left(t-x-\frac{1}{2}\right)f\left(t\right)dt+{\int }_0^x(x-t)f\left(t\right)dt.\hfill \end{array}$$

5. 

For $m=1$, we set up the vectors

$$\begin{array}{ccc}& & S=\left(8x^2\right),G=\left(3x+1\right),\hfill \\ & & \mathsf{\Phi }\left(A_{1}u\right)={\int }_0^{1}t{u}^{\prime }\left(t\right)dt,\mathsf{\Psi }\left(Au\right)={\int }_0^1{t}^2{u}^{‴}\left(t\right)dt.\hfill \end{array}$$

6. 

We compute

$$detW=det[I_m-\mathsf{\Psi }\left(G\right)]=-\frac{1}{12}\ne 0.$$

(23)

7. 

By utilizing (13), we find

$$P=\left(\frac{10x^4-144x^3-144x^2+340x-31}{15}\right).$$

(24)

8. 

Finally, we compute

$$detV=det[I_m-\mathsf{\Phi }\left(P\right)]=-\frac{197}{150}\ne 0.$$

(25)

9. 

From (22)–(25) it is implied that all requirements of Theorem 2 are satisfied and therefore the given boundary value problem (20), (21) can be factorized and its unique solution may be obtained by substituting into (17).

As an illustration, for $f\left(x\right)=2x^2-6x+4$ the unique solution turns out to be

$$u\left(x\right)=x^3-\frac{3}{2}{x}^2+\frac{1}{2}.$$

Example 2.

Let now the partial integro-differential equation

$$u_{xy}-(x+y){\int }_0^1{\int }_0^{1}t{u}_t(t,s)dtds-3x^3{\int }_0^1{\int }_0^1{s}^2{u}_{ts}(t,s)dtds=f(x,y),$$

(26)

along with the boundary conditions

$$u_x(x,0)=0,u(0,y)=(y+1){\int }_0^{1}u(t,y)dt,$$

(27)

where $f(x,y)\in C\left(\mathsf{\Omega }\right)$ and $\mathsf{\Omega }=[0,1]\times [0,1]$.

Following the steps described in Listing 1, we have:

1. 

Take $X=C\left(\mathsf{\Omega }\right)$ and define the operator $B:C\left(\mathsf{\Omega }\right)\to C\left(\mathsf{\Omega }\right)$ as

$$\begin{array}{ccc}\hfill Bu& =& u_{xy}-(x+y){\int }_0^1{\int }_0^{1}t{u}_t(t,s)dtds-3x^3{\int }_0^1{\int }_0^1{s}^2{u}_{ts}(t,s)dtds,\hfill \\ \hfill D\left(B\right)& =& \left\{u\in C\left(\mathsf{\Omega }\right):u_x,u_{xy}\in C\left(\mathsf{\Omega }\right),u_x(x,0)=0,u(0,y)=(y+1){\int }_0^{1}u(t,y)dt\right\}.\hfill \end{array}$$

2. 

Next, we define the operators $A,A_1:C\left(\mathsf{\Omega }\right)\to C\left(\mathsf{\Omega }\right)$ of order $n=2$ and $n_1=1$, respectively, by

$$\begin{array}{ccc}\hfill Au& =& u_{xy},D\left(A\right)=D\left(B\right),\hfill \\ \hfill A_{1}u& =& u_x,D\left(A_1\right)=\left\{u\in C\left(\mathsf{\Omega }\right):u_x\in C\left(\mathsf{\Omega }\right),u(0,y)=(y+1){\int }_0^{1}u(t,y)dt\right\}.\hfill \end{array}$$

3. 

The operator $A_2:C\left(\mathsf{\Omega }\right)\to C\left(\mathsf{\Omega }\right)$ of order $n_2=1$ is determined by the requirement that

$$\begin{array}{ccc}Au& =& u_{xy}=A_2{A}_{1}u=A_2\left(u_x\right)=A_{2}v=v_y,v=u_x(x,y),\hfill \\ D\left(A_2{A}_1\right)& =& \left\{u\in D\left(A_1\right):A_{1}u=u_x=v\in D\left(A_2\right)\right\}\hfill \\ & =& \left\{u\in C\left(\mathsf{\Omega }\right):u_x\in C\left(\mathsf{\Omega }\right),u(0,y)=(y+1){\int }_0^{1}u(t,y)dt,u_x=v\in D\left(A_2\right)\right\}\hfill \\ & =& D\left(B\right),\hfill \end{array}$$

(28)

which yields

$$A_{2}v=v_y,D\left(A_2\right)=\left\{v\in C\left(\mathsf{\Omega }\right):v_y\in C\left(\mathsf{\Omega }\right),v(x,0)=0\right\}.$$

4. 

The operator $A_1$ is correct if and only if $y\ne 0$ in which case its inverse is given by

$$A_1^{-1}f={\int }_0^{x}f(t,y)dt-\frac{y+1}{y}{\int }_0^1(1-t)f(t,y)dt,$$

and the operator $A_2$ is correct with its inverse given by

$$A_2^{-1}f={\int }_0^{y}f(x,s)ds.$$

5. 

We take $m=1$ and set up the vectors

$$\begin{array}{ccc}\hfill S& =& (x+y),G=\left(3x^3\right)\hfill \\ \hfill \mathsf{\Phi }\left(A_{1}u\right)& =& {\int }_0^1{\int }_0^{1}t{u}_t(t,s)dtds,\mathsf{\Psi }\left(Au\right)={\int }_0^1{\int }_0^1{y}^2{u}_{ts}(t,s)dtds.\hfill \end{array}$$

6. 

We compute

$$detW=1-\mathsf{\Psi }\left(G\right)=\frac{3}{4}\ne 0.$$

(29)

7. 

Consequently, from (13) we obtain

$$P=\left(xy+\frac{y^2}{2}+\frac{5}{3}{x}^{3}y\right).$$

(30)

8. 

We also compute

$$detV=1-\mathsf{\Phi }\left(P\right)=\frac{7}{12}\ne 0.$$

(31)

9. 

From (28)–(31) it follows that all requirements of Theorem 2 are fulfilled and therefore the given boundary value problem (26), (27) can be factorized and its unique solution can be obtained by substituting into (17).

By way of illustration, let $f(x,y)=15x^3-2x-2y$. Then the unique solution of the boundary value problem (26), (27) is

$$u(x,y)=5x^{4}y-y-1.$$

5. Discussion

Factorization techniques have been known for long time and have been used extensively for studying and solving linear and nonlinear differential equations. They are very efficacious and useful whenever they can be applied.

In this paper, an attempt was made to extend the applicability of factorization methods to boundary value problems for a category of linear Fredholm type integro-differential equations with nonlocal boundary conditions. It is found that factoring the corresponding integro-differential operator is more difficult than the decomposition of analogous differential operators. It requires the solution of two lower-order integro-differential equations.

The proposed technique was tested and found to be very successful in solving exact boundary value problems for both ordinary and partial integro-differential equations. As it is the case with factorization methods, its application is limited to certain classes of integro-differential equations.

The main contribution of the paper is Theorems 2 and 3, and the procedure in Listing Section 3 which can be implemented to any software of symbolic computations.

The method can be advanced and extended further to include more types of integro-differential operators.