# Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams
## 2 Closed-Form Solution of Volterra-Fredholm Integro-Differential Equations  
Let \(X=C[0,L]\), \(L\in\mathbb{R}^{+}\), and \(A:X\to X\) be an \(n\)th order linear differential operator of the form  
\[Au=\sum_{i=0}^{n}a_{i}u^{(n-i)}(x),\quad\mathcal{D}(A)=\{u\in X_{n}:\Phi(u)= \mathbf{0}\}, \tag{1}\]  
where \(n\in\mathbb{N}\), \(a_{i}\), \(i=0,1,\ldots,n\), are real constants with \(a_{0}\
eq 0\), \(X_{n}=C^{n}[0,L]\), \(X_{0}=X\), \(u=u(x)\in X_{n}\), \(u^{(i)}(x)=\frac{\phi_{i}u}{dx^{i}}\), \(i=1,2,\ldots,n\),  
\[\Phi(u)=\left(\begin{array}{c}\Phi_{1}(u)\\ \Phi_{2}(u)\\ \vdots\\ \Phi_{n}(u)\end{array}\right),\quad\Phi\in[X_{n-1}^{*}]^{n},\quad\Phi_{i}\in X _{n-1}^{*},\ i=1,2,\ldots,n, \tag{2}\]  
is a column vector of linear functionals that describe the specified boundary conditions, and \(\mathbf{0}\) denotes the zero column vector.  
Let \(K:X\to X\) be the linear Volterra integral operator of convolution type  
\[Ku=\sum_{i=0}^{n}\int_{0}^{x}k_{i}(x-t)u^{(n-i)}(t)dt, \tag{3}\]  
where the kernels \(k_{i}(x)\in X\), \(i=0,1,\ldots,n\).  
Let the Fredholm-type functionals  
\[\int_{0}^{L}\bar{k}_{j}(x,t)Au(t)dt,\quad j=1,2,\ldots,m, \tag{4}\]  
where the kernels \(\bar{k}_{j}(x,t)\in X\times X\) are assumed to be separable, i.e.,  
\[\bar{k}_{j}(x,t)=g_{j}(x)h_{j}(t),\quad g_{j}=g_{j}(x),\ h_{j}=h_{j}(t)\in X, \quad j=1,2,\ldots,m. \tag{5}\]  
Let the row vector of functions  
\[g=\big{(}\begin{array}{cccc}g_{1}&g_{2}&\cdots&g_{m}\end{array}\big{)},\quad g \in X^{m}, \tag{6}\]  
and the column vector of functionals  
\[\Psi(Au)=\left(\begin{array}{c}\Psi_{1}(Au)\\ \Psi_{2}(Au)\\ \vdots\\ \Psi_{m}(Au)\end{array}\right),\quad\Psi_{j}(Au)=\int_{0}^{L}h_{j}(t)Au(t)dt, \quad j=1,2,\ldots,m, \tag{7}\]  
where \(\Psi\in[X^{*}]^{m}\) and \(\Psi_{j}\in X^{*}\), \(j=1,2,\ldots,m\).  
Consider the linear Volterra-Fredholm type integro-differential operator \(T:X\to X\) defined by  
\[Tu = Au+Ku-\sum_{j=1}^{m}g_{j}\Psi_{j}(Au)=Au+Ku-g\Psi(Au),\] \[\mathcal{D}(T) = \mathcal{D}(A)=\{u\in X_{n}:\Phi(u)=\mathbf{0}\}, \tag{8}\]  
and the Volterra-Fredholm integro-differential boundary value problem  
\[VFIDBVP:\quad Tu=f,\quad f=f(x)\in X. \tag{9}\]  
Let the \(m\times m\) matrix  
\[\Psi(g)=\left(\begin{array}{cccc}\Psi_{1}(g_{1})&\Psi_{1}(g_{2})&\cdots&\Psi _{1}(g_{m})\\ \Psi_{2}(g_{1})&\Psi_{2}(g_{2})&\cdots&\Psi_{2}(g_{m})\\ \vdots&\vdots&\ddots&\vdots\\ \Psi_{m}(g_{1})&\Psi_{m}(g_{2})&\cdots&\Psi_{m}(g_{m})\end{array}\right),\]  
where the element \(\Psi_{i}(g_{j})\) is the value of the functional \(\Psi_{i}\) on the element \(g_{j}\), and \(I_{m}\) denotes the \(m\times m\) identity matrix.  
The criteria for the existence of a unique solution of the VFIDBVP in (9) and a formula for its symbolic calculation in an exact closed form are given in [39] where the following theorem has been proved.  
Theorem 1: _Let the operator \(T:X\to X\) be defined as in (8). Assume that the Volterra integro-differential operator \(D:X\to X\) defined by_  
\[Du=(A+K)u,\quad\mathcal{D}(D)=\mathcal{D}(A), \tag{10}\]  
_is bijective on \(X\) and its inverse is denoted by \(D^{-1}=(A+K)^{-1}\). Then, the operator \(T\) is bijective, precisely it is injective if and only if_  
\[\det W=\det\Bigl{[}I_{m}-\Psi\Bigl{(}AD^{-1}g\Bigr{)}\Bigr{]}\
eq 0, \tag{11}\]and in this case, the unique solution to the boundary value problem_  
\[Tu=f,\quad\text{for all functions}\quad f\in X, \tag{12}\]  
_is given by the formula_  
\[u = T^{-1}f \tag{13}\] \[= D^{-1}f+D^{-1}gW^{-1}\mathbf{v}\Big{(}AD^{-1}f\Big{)}.\]