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# An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

by
Efthimios Providas
Department of Environmental Sciences, Gaiopolis Campus, University of Thessaly, 41500 Larissa, Greece
Algorithms 2022, 15(6), 203; https://doi.org/10.3390/a15060203
Submission received: 22 May 2022 / Revised: 5 June 2022 / Accepted: 8 June 2022 / Published: 12 June 2022
(This article belongs to the Section Algorithms for Multidisciplinary Applications)

## Abstract

:
In this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering.
MSC:
45A05; 45G10; 45B05; 45D05; 45P05

## 1. Introduction

Mathematical modeling in physics, life sciences and engineering very often leads to integral equations. Examples of specific fields include electromagnetism, biology, population dynamics, genetics, epidemiology, heat transfer, elasticity, viscoelasticity, hydrodynamics and telecommunications [1,2,3,4]. Integral equations (IE) may have an integral term with variable limits (Volterra IE), fixed limits (Fredholm IE) or both (Volterra–Fredholm IE), with the unknown function appearing only in the integrand (IE of the first kind) or in the integrand as well as outside the integral (IE of the second kind). The Volterra–Fredholm integral equations (VFIE) appear in two forms [5], namely in a form where Volterra and Fredholm type integrals are disjoint [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90] and in a mixed form where these two integrals are together in a multiple integral, (see [91,92,93,94,95,96,97,98,99,100,101,102,103] and the references therein). As an example of the two types of one-dimensional linear VFIE of the second kind, we have
$y ( x ) = f ( x ) + ∫ a x k 1 ( x , t ) y ( t ) d t + ∫ a b k 2 ( x , t ) y ( t ) d t , a ≤ x ≤ b ,$
and
$y ( x ) = f ( x ) + ∫ a x ∫ a b k ( s , t ) y ( t ) d t d s , a ≤ x ≤ b ,$
respectively, where $f ( x )$, $k i ( x , t ) , i = 1 , 2 ,$ and $k ( x , t )$ are given continuous functions and $y ( x )$ is the unknown function.
Due to their complexity, VFIE are mainly solved by numerical methods. In recent decades, many different computational methods have been developed to solve approximately both linear and nonlinear VFIE. Taylor series expansion methods have been developed in  [5,10,12,15,30]. The Adomian decomposition method [5,95] and the modified decomposition method [5,17,49,96] are very powerful methods and, in some cases, can give the exact solution. Collocation methods are very popular. They are based on splines [21,29,92,93], general approximate functions [22,82], Chebyshev polynomials [16], shifted Chebyshev polynomials [81], Bernstein polynomials [34,63], Chelyshkov Polynomials [36], Lagrange polynomials [42], Taylor polynomials [47], Fibonacci polynomials [57], Bell polynomials [65], first Boubaker polynomials [60], Müntz–Legendre polynomials [70], generalized Lucas polynomials [87], Jacobi polynomials [89], block-pulse functions [27], hybrid of block-pulse functions and Lagrange polynomials [26], hybrid block-pulse function and Taylor polynomials [41] (see also [73]), block-pulse functions and Bernoulli polynomials [45], hybrid block-pulse functions and Bernstein polynomials [62], Haar wavelets [4,66], rationalized Haar functions [19], Legendre wavelets [13,97], triangular functions [23,100], fuzzy transforms [32], Sinc function [37,40], radial basis functions [35], pseudospectral integration matrices [54] and shifted piecewise cosine basis functions [75]. Galerkin methods are also popular and are commonly used in conjunction with general approximate functions [22,82], Legendre polynomials [53], Bernstein polynomials [69], coiflet-type wavelets [46] and Alpert’s multiwavelet bases [80]. Quadrature methods [43,64], least squares approximation methods [31,44,48,103], homotopy analysis methods [24,79], homotopy perturbation methods [18,25,38] and modified homotopy perturbation methods [99] have also been developed. Typical examples of iterative approximate methods are the Picard iteration method [51,101], Runge–Kutta method and block by block method [76], optimal perturbation iteration method [77], Dzyadyk’s iterative approximation method [94] and Mann iteration procedure [102]. Fixed point methods based on Schauder bases [21,29,33,98], Schauder basis in an adequate Banach space [58], rationalized Haar wavelet [68], and fixed point methods in extended b-metric space [71] have been used successfully. Approximate methods using block-pulse functions [52,61,67], Bernstein polynomials [59], Haar wavelets [4], hyperbolic basis functions [84] and Hosoya polynomials [85] have also been presented. Other successful methods that have been reported are the reproducing kernel methods [14,78,88], Tau methods [50,55], modified hat functions method [56], optimal control method [28], scaling function interpolation wavelet method [39], hybrid contractive mapping and parameter continuation method [72], sinusoidal basis functions and a neural network approach [74]. Finally, existence and uniqueness results are considered in [6,7,8,9,11,20,83,91].
The above analysis shows that there are no procedures for the exact closed-form solution of VFIE. It is understandable that such methods are very difficult to develop due to the nature of the problem, especially in the nonlinear case, and whether they exist they will relate to specific categories of VFIE. Methods for solving Volterra IE and Fredholm IE exactly are given in [104], which describes the method for integral transforms, the method for differentiation and the direct method for integral equations in both the Urysohn form and the Hammerstein form. Motivated by this work and based on our previous experience with the closed-form solution of the Fredholm-type integro-differential equations [105,106,107,108], we consider the VFIE of the second kind in the form
$y ( x ) = f ( x ) + ∑ i = 1 n ∫ 0 x k i ( x , t ) y ( t ) d t + ∑ j = 1 m ∫ 0 b k ¯ j x , t , y ( t ) d t , 0 ≤ x ≤ b ,$
where
$k ¯ j x , t , y ( t ) = q ¯ j ( x , t ) φ ¯ j ( t , y ( t ) ) , j = 1 , 2 , … , m .$
The kernels $k i ( x , t ) , i = 1 , 2 , … , n ,$ and $q ¯ j ( x , t ) , j = 1 , 2 , … , m ,$ the functions $φ ¯ j ( t , y ( t ) ) ,$$j = 1 , 2 , … , m ,$ and the input function $f ( x )$ are given continuous functions, and $y ( x )$ is the unknown function. Equation (1) with a kernel of the form (2) is called the Hammerstein equation of the second kind. When all functions $φ ¯ j ( t , y ) , j = 1 , 2 , … , m ,$ are linear in $y ,$ then Equation (1) is a linear integral equation. We formulate (1) in a convenient operator form and derive its closed-form solution when the inverse of the associated integral Volterra operator of the second kind is available in explicit closed form, and the kernels $q ¯ j ( x , t ) , j = 1 , 2 , … , m ,$ are degenerate. In the linear case, i.e., when all functions $φ ¯ j ( t , y ) , j = 1 , 2 , … , m ,$ are linear in y, we establish results of the existence and uniqueness. We elaborate on the Volterra operators of convolution type (CVFIE) and their inversion by the Laplace transform method. A complete algorithm for the proposed solution method is provided.
The rest of the paper is organized as follows: In Section 2, an operator formulation of the problem is given, and the solution technique is explained. In Section 3, the case of convolution-type kernels is studied, and an algorithm is presented. In Section 4, we solve various illustrative examples to show the efficiency of the method. Lastly, some conclusions are given in Section 5.

## 2. Direct Operator Method for Solving Linear and Nonlinear VFIE

In the space $X = C [ 0 , b ]$, $b ∈ R +$, let $K : X → X$ be the linear Volterra integral operator defined by
$K y ( x ) = ∑ i = 1 n ∫ 0 x k i ( x , t ) y ( t ) d t ,$
where each of the kernels $k i ( x , t ) ∈ X × X , i = 1 , … , n ,$ and $y ( x ) ∈ X$.
Let the Fredholm type integrals
$∫ 0 b k ¯ j x , t , y ( t ) d t = ∫ 0 b q ¯ j ( x , t ) φ ¯ j ( t , y ( t ) ) d t = g j ( x ) ∫ 0 b h j ( t ) φ ¯ j ( t , y ( t ) ) d t , j = 1 , 2 , … , m ,$
where the kernels $q ¯ j ( x , t )$ are supposed to be degenerate in the form $q ¯ j ( x , t ) = g j ( x ) h j ( t )$ and $g j ( x ) , h j ( t ) ∈ X$. Define the vectors
$g = g 1 g 2 ⋯ g m , g j = g j ( x ) , j = 1 , 2 , … , m ,$
$Φ ( y ) = Φ 1 ( y ) Φ 2 ( y ) ⋮ Φ m ( y ) , Φ j ( y ) = ∫ 0 b h j ( t ) φ ¯ j ( t , y ( t ) ) d t , j = 1 , 2 , … , m ,$
and note that $g ∈ X m$, and $Φ$ is vector of m nonlinear functionals $Φ j$ defined on X.
By using (3)–(6) define the Volterra–Fredholm integral operator $I : X → X$ as
$I y = y − K y − ∑ j = 1 m g j Φ j ( y ) = y − K y − g Φ ( y ) , y = y ( x ) ∈ X ,$
and write the VFIE in (1) in the symbolic form
$I y = f , f = f ( x ) ∈ X .$
Let $Φ ( g )$ be the $m × m$ matrix
$Φ ( g ) = Φ 1 ( g 1 ) ⋯ Φ 1 ( g m ) ⋮ ⋱ ⋮ Φ m ( g 1 ) ⋯ Φ m ( g m ) ,$
where the element $Φ i ( g j )$ is the value of the functional $Φ i$ on the element $g j$, $I m$ the $m × m$ identity matrix, $c$ a $m × 1$ column constant vector and $0$ the $m × 1$ column zero vector.

#### 2.1. Linear VFIE

First, we consider the case of linear VFIE where without loss of generality we assume that $φ ¯ j ( t , y ( t ) ) ≡ y ( t ) , j = 1 , 2 , … , m$. In this case, the Fredholm type functionals $Φ j ( y )$ in (6) become
$Φ j ( y ) = ∫ 0 b h j ( t ) y ( t ) d t , j = 1 , 2 , … , m ,$
which are bounded linear functionals, and the vector $Φ ∈ [ X ∗ ] m$ where $X ∗$ is the space of all bounded linear functionals on X. Moreover, for a vector of functions g, such as in (5), and a constant vector $c$, the following relation holds
$Φ ( g c ) = Φ ( g ) c .$
Theorem 1
(Linear VFIE). Let the linear Volterra–Fredholm integral operator of the second kind $I : X → X$ be defined by
$I y = J y − g Φ ( y ) , y = y ( x ) ∈ X ,$
where the Volterra-type integral operator of the second kind $J : X → X$ is defined as
$J y = y − K y , y = y ( x ) ∈ X ,$
and the operator $K$ as in (3), the vector of functions g as in (5) and Φ is a vector of linear bounded functionals of the kind (7). If the integral operator $J$ is bijective on X, and its inverse is $J − 1$, then the operator $I$ is injective if and only if
$det W = det I m − Φ J − 1 g ≠ 0 .$
Furthermore, the unique solution of the linear VFIE
is given by
$y = I − 1 f = J − 1 f + J − 1 g W − 1 Φ J − 1 f .$
Proof.
Let us assume that $det W ≠ 0$ and take an element $y ∈ ker I$. Then
$I y = J y − g Φ ( y ) = 0 ,$
and since the operator $J$ is bijective, we have
$y = J − 1 g Φ ( y ) .$
Acting by the vector $Φ$ on both sides of (14) and by using (8), we get
$Φ ( y ) = Φ J − 1 g Φ ( y ) = Φ J − 1 g Φ ( y ) ,$
and hence
$I m − Φ J − 1 g Φ ( y ) = W Φ ( y ) = 0 .$
Since $det W ≠ 0$, we have $Φ ( y ) = 0$. Then, from (14), it follows that $y = 0$, i.e., $ker I = { 0 }$, which implies that the operator $I$ is injective. Conversely, we assume $I$ is injective, and we will show that $det W ≠ 0$, or, equivalently, we assume $det W = 0$, and we will prove that $I$ is not injective. In this case, there exists a nonzero constant vector $c$ such that $W c = 0$. Let the element $y 0 = J − 1 g c ∈ X$. Note that $y 0 ≠ 0$ since $y 0 = 0$, we have
$W c = I m − Φ J − 1 g c = c − Φ J − 1 g c = c − Φ J − 1 g c = c − Φ y 0 = c = 0 ,$
which contradicts the assumption that $c$ is nonzero. Then
$I y 0 = J y 0 − g Φ ( y 0 ) = g c − g Φ ( J − 1 g c ) = g c − g Φ ( J − 1 g ) c = g I m − Φ J − 1 g c = g W c = 0 .$
This means that $ker I ≠ { 0 }$, and, therefore, $I$ is not injective.
Let now the VFIE in (12), viz.
$I y = J y − g Φ ( y ) = f , f ∈ X .$
By applyingthe operator $J − 1$ on (15), we get
$y = J − 1 f + J − 1 g Φ ( y ) .$
Acting as aboveby the vector $Φ$, we have
$Φ ( y ) = Φ J − 1 f + Φ J − 1 g Φ ( y ) ,$
and hence
$I m − Φ J − 1 g Φ ( y ) = Φ J − 1 f , Φ ( y ) = I m − Φ J − 1 g − 1 Φ J − 1 f = W − 1 Φ J − 1 f .$
Substitution of (17) into (16) yields
$y = J − 1 f + J − 1 g W − 1 Φ J − 1 f ,$
which is the solution of the linear VFIE in (12) for every $f ∈ X$.    □

#### 2.2. Nonlinear VFIE

Consider now the nonlinear VFIE in the Hammerstein form (1) where the Fredholm type functionals $Φ j ( y )$ are nonlinear and are given in (6).
Theorem 2
(Nonlinear VFIE). Let the nonlinear Volterra–Fredholm integral operator of the second kind $I : X → X$ be defined by
$I y = J y − g Φ ( y ) , y = y ( x ) ∈ X ,$
where the Volterra-type integral operator of the second kind $J : X → X$ is defined as
$J y = y − K y , y = y ( x ) ∈ X ,$
and the operator $K$ as in (3), and the vector of functions g and the vector of nonlinear functionals Φ as in (5) and (6), respectively. If the integral operator $J$ is bijective on X, and its inverse is denoted by $J − 1$, then the exact solution of the nonlinear VFIE
$I y = f , f = f ( x ) ∈ X ,$
is given by
$y = J − 1 f + J − 1 g a ∗ ,$
for every vector $a ∗ = Φ ( y )$, which is a solution of the nonlinear algebraic (transcendental) system of the m equations
$a = Φ J − 1 f + J − 1 g a .$
Proof.
Applying the inverse operator $J − 1$ to the nonlinear VFIE in (20), we get
$y = J − 1 f + J − 1 g Φ ( y ) .$
Acting then by the vector $Φ$ on both sides, we have
$Φ ( y ) = Φ J − 1 f + J − 1 g Φ ( y ) ,$
where it is reminded that the elements of the vector $Φ$ are nonlinear functionals, and no linear operations are allowed. By defining $a = Φ ( y )$, we obtain the nonlinear algebraic (transcendental) system
$a = Φ J − 1 f + J − 1 g a .$
Let $a ∗$ be a solution of this system. Substitution then of $a ∗$ into (22) yields (21). The solution $y ( x )$ satisfying $a ∗ = Φ ( y )$ is a solution of the nonlinear VFIE (21).    □

## 3. Solving VFIE of Convolution Type

The method described in the previous section requires the existence and construction in closed form of the inverse operator $J − 1$ of the linear Volterra operator of the second kind $J$. Finding the exact inverse is not an easy task, and in many cases it is not possible. Polyanin and Manzhirov [104] provide the closed-form solution for several classes of linear Volterra integral equations (VIE) of the second kind. Here, we distinguish and deal with the special case of the VIE of convolution type that can be explicitly solved by applying the Laplace transform method.
It is well known that the Laplace transform of a function $f ( x )$ defined for all $x ≥ 0$ is a function $F ( s )$ of the variable s defined by
$F ( s ) = L ( f ) = ∫ 0 ∞ e − s x f ( x ) d x ,$
where $L$ denotes the Laplace transform operator, which is linear and injective. Suppose $f ( x )$ is continuous and of exponential order; that is, there exist real constants $γ$ and M such that $| f ( x ) | ≤ M e γ x$ for all $x ≥ 0$. Then the Laplace transform of f exists for all $s > γ$. The inverse of the Laplace transform is $f ( x ) = L − 1 ( F )$. The convolution of two functions of exponential order $f , g$ is defined by
$( f ∗ g ) ( x ) = ∫ 0 x f ( x − t ) g ( t ) d t ,$
and $L ( f ∗ g ) = L ( f ) L ( g )$ (see for example [109,110]).
Let the linear Volterra integral operator $K$ in (3) be of convolution type, i.e.,
$K y ( x ) = ∑ i = 1 n ∫ 0 x k i ( x − t ) y ( t ) d t = ∑ i = 1 n k i ∗ y ( x ) ,$
and let the convolution-type Volterra integral equation (CVIE) of the second kind
$J y ( x ) = y ( x ) − K y ( x ) = y ( x ) − ∑ i = 1 n k i ∗ y ( x ) = r ( x ) , r ∈ X .$
By applying the Laplace transform operator on both sides of (24) and utilizing the convolution theorem, we get
$L J y = L y − ∑ i = 1 n k i ∗ y = Y ( s ) − Y ( s ) ∑ i = 1 n K i ( s ) = R ( s ) ,$
where $Y ( s ) = L y$, $K i ( s ) = L k i , i = 1 , 2 , … , m ,$ and $R ( s ) = L r$. Multiplication by the transfer function
$Q ( s ) = 1 1 − K 1 ( s ) + K 2 ( s ) + ⋯ + K n ( s ) ,$
yields
$Y ( s ) = R ( s ) Q ( s ) .$
Taking the inverse Laplace transform, we obtain the solution of (24), namely
$y = J − 1 r = L − 1 R ( s ) Q ( s ) .$
Since the Equation (25) holds for every $r ( x ) ∈ X$, it also implies that the operator $J$ is bijective.
To summarize and facilitate the implementation of the methods presented in the previous section and here, we provide the following algorithm in Algorithm 1.
 Algorithm 1: Algorithm for the implementation of the Laplace transform to construct the inverse Volterra operator $J − 1$ and the Theorems 1 and 2 to obtain the exact solution of the linear (12) and nonlinear VFIE (20), respectively. input     $k 1 ( x ) , k 2 ( x ) , … , k n ( x ) ,$     $g 1 ( x ) , g 2 ( x ) , … , g m ( x ) , h 1 ( x ) , h 2 ( x ) , … , h m ( x ) , b ,$          $f ( x )$  compute     $K i ( s ) : = L k i , i = 1 , 2 , … , n$     $Q ( s ) : = 1 1 − K 1 ( s ) + K 2 ( s ) + ⋯ + K n ( s )$     $G j ( s ) : = L g j , j = 1 , 2 , … , m$         $J − 1 g j ( x ) : = L − 1 G j ( s ) Q ( s ) , j = 1 , 2 , … , m$     $J − 1 g : = J − 1 g 1 ( x ) ⋯ J − 1 g m ( x )$     $F ( s ) : = L { f }$     $J − 1 f : = L − 1 { F ( s ) Q ( s ) }$     linear case $φ ¯ j ( t , y ( t ) ) ≡ y ( t )$:       $Φ i J − 1 g j : = ∫ a b h i ( t ) J − 1 g j ( t ) d t , i , j = 1 , 2 , … , m$       $Φ J − 1 g : = Φ 1 J − 1 g 1 ⋯ Φ 1 J − 1 g m ⋮ ⋱ ⋮ Φ m J − 1 g 1 ⋯ Φ m J − 1 g m$       $W : = I m − Φ J − 1 g$       if$det W ≠ 0$  compute       $Φ i J − 1 f : = ∫ a b h i ( t ) J − 1 f ( t ) d t , i = 1 , 2 , … , m$        $Φ J − 1 f : = Φ 1 J − 1 f ⋮ Φ m J − 1 f$       $y : = J − 1 f + J − 1 g W − 1 Φ J − 1 f$       print $y ( x )$       else       print ’There is no unique solution’       end     nonlinear case $define φ ¯ j ( t , y ( t ) )$:       $a : = col a 1 ⋯ a m$       $v ( t ) : = J − 1 f + J − 1 g a$       $Φ i v : = ∫ 0 b h i ( t ) φ ¯ j ( t , y ( t ) ) d t , i = 1 , 2 , … , m$       $Φ v : = Φ 1 v ⋮ Φ m v$       solve the system       $a = Φ v$       for every solution $a ∗$ compute       $y : = J − 1 f + J − 1 g a ∗$       if y satisfies  $a ∗ − Φ ( y ) = 0$  print the solution  $y ( x )$       end  end

## 4. Examples

In this section, we solve three linear and three nonlinear problems with VFIE of the convolution type to show the capabilities, ease of use and cost-effectiveness of the proposed method.

#### 4.1. Example 1

The following illustrative example of linear CVFIE is solved in [5] by means of the Taylor series solution method and in [65] numerically by means of a collocation method based on Bell polynomials,
$u ( x ) = − x 4 − x 3 + 12 x 2 − x − 5 + ∫ 0 x ( x − t ) u ( t ) d t + ∫ 0 1 ( x + t ) u ( t ) d t , 0 ≤ x ≤ 1 .$
Here, we solve it using the proposed direct operator method. Since the kernel of the Fredholm integral operator is degenerate, we write (26) in the form
$u ( x ) − ∫ 0 x ( x − t ) u ( t ) d t − x ∫ 0 1 u ( t ) d t − ∫ 0 1 t u ( t ) d t = − x 4 − x 3 + 12 x 2 − x − 5 .$
Juxtaposing expression (27) with (12) in Theorem 1, we take $n = 1$, $m = 2$, $b = 1$ and
$k 1 ( x ) = x , g 1 ( x ) = x , g 2 ( x ) = 1 , h 1 ( t ) = 1 , h 2 ( t ) = t , φ ¯ 1 ( t , u ( t ) ) = φ ¯ 2 ( t , u ( t ) ) = u ( t ) , f ( x ) = − x 4 − x 3 + 12 x 2 − x − 5 .$
The convolution kernel $k 1 ( x )$ and the functions $g 1 ( x ) , g 2 ( x )$ and $f ( x )$ are continuous and of exponential order, and so, their Laplace transforms $L k 1 ( x ) , L g 1 ( x ) , L g 2 ( x )$ and $L f ( x )$, respectively, exist. As a result, the Volterra integral operator of the second kind $J$ is invertable.
When entering the functions (28) in the Algorithm 1 and executing, we get the exact solution
$u ( x ) = 12 x 2 + 6 x .$

#### 4.2. Example 2

Let the linear CVFIE of the second kind
$u ( x ) = e x + 1 − 4 e x + x 2 + ∫ 0 x e x − t u ( t ) d t − 3 ∫ 0 1 e x + t u ( t ) d t . 0 ≤ x ≤ 1 .$
Note that the kernel of the Fredholm integral operator in (29) is degenerate, and therefore Equation (29) can be written as follows
$u ( x ) − ∫ 0 x e x − t u ( t ) d t + 3 e x ∫ 0 1 e t u ( t ) d t = e x + 1 − 4 e x + x 2 .$
Comparing (30) with (12) in Theorem 1, we set $n = m = 1$, $b = 1$ and
$k 1 ( x ) = e x , g 1 ( x ) = − 3 e x , h 1 ( t ) = e t , φ ¯ 1 ( t , u ( t ) ) = u ( t ) , f ( x ) = e x + 1 − 4 e x + x 2 ,$
The convolution kernel $k 1 ( x )$ and the functions $g 1 ( x )$ and $f ( x )$ are continuous and of exponential order, and therefore their corresponding Laplace transforms exist, which in turn implies that the Volterra integral operator of the second kind $J$ is invertable. Then, by giving the functions (31) as input in Algorithm 1, and after performing the calculations, we get the exact solution
$u ( x ) = 1 4 e 2 x − 2 + 2 x 2 − 2 x − 1 .$

#### 4.3. Example 3

The Fredholm integral equation of the second kind
$u ( x ) = f ( x ) + 1 2 ∫ 0 1 e − | x − t | u ( t ) d t , 0 ≤ x ≤ 1 ,$
appears in the one-dimensional transport process [111] and the bending analysis of nonlocal integral models of Euler–Bernoulli nanobeams [112,113,114].
This equation, by removing the modulus in the integrand, can be converted to a Volterra–Fredholm integral equation of convolution type, namely
$u ( x ) = f ( x ) + 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t + ∫ x 1 e x − t u ( t ) d t = f ( x ) + 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t − ∫ 0 x e x − t u ( t ) d t + ∫ 0 1 e x − t u ( t ) d t = f ( x ) + 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t − ∫ 0 x e x − t u ( t ) d t + 1 2 ∫ 0 1 e x − t u ( t ) d t = f ( x ) + 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t − ∫ 0 x e x − t u ( t ) d t + 1 2 e x ∫ 0 1 e − t u ( t ) d t$
or
$u ( x ) − 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t − ∫ 0 x e x − t u ( t ) d t − 1 2 e x ∫ 0 1 e − t u ( t ) d t = f ( x ) .$
This linear CVFIE can now be solved by the direct operator method presented in previous sections. In particular, comparing (33) with (12) in Theorem 1 it is natural to take $n = 2$, $m = 1$, $b = 1$ and
$K u ( x ) = 1 2 ∫ 0 x e − ( x − t ) u ( t ) d t − ∫ 0 x e x − t u ( t ) d t ,$
which is a convolution Volterra integral operator of the form (23) with
$k 1 ( x ) = 1 2 e − x , k 2 ( x ) = − 1 2 e x .$
$g 1 ( x ) = 1 2 e x , h 1 ( t ) = e − t , φ ¯ 1 ( t , u ( t ) ) = u ( t ) ,$
and let
$f ( x ) = 1 2 e − x + e − ( 1 − x ) ,$
as in [111].
The functions $k 1 ( x ) , k 2 ( x )$, $g 1 ( x )$ and $f ( x )$ are continuous and of exponential order, and therefore the Volterra integral operator of the second kind $J$ is invertable. By entering the functions in (34)–(36) in the Algorithm 1 and executing, we get the exact solution of the CVFIE (33), viz.
$u ( x ) = 1 ,$
which is the solution of the given Fredholm integral Equation (32) with $f ( x )$ as in (36).
In Table 1, we give the exact solution corresponding to three other types of the input function $f ( x )$.

#### 4.4. Example 4

Consider the nonlinear CVFIE of the second kind
$u ( x ) = − x 4 + ∫ 0 x sin ( x − t ) u ( t ) d t − 1 10 ∫ 0 2 x t u 2 ( t ) d t , 0 ≤ x ≤ 2 ,$
where the Fredholm integral operator is nonlinear and degenerate. We write Equation (37) in the form
$u ( x ) − ∫ 0 x sin ( x − t ) u ( t ) d t + x 10 ∫ 0 2 t u 2 ( t ) d t = − x 4 .$
Comparing (38) with (20) in Theorem 2, we take $n = m = 1$, $b = 2$ and
$k 1 ( x ) = sin x , g 1 ( x ) = − x 10 , h 1 ( t ) = t , φ ¯ 1 ( t , u ( t ) ) = u 2 ( t ) , f ( x ) = − x 4 .$
Since the convolution kernel $k 1 ( x )$ and the functions $g 1 ( x )$ and $f ( x )$ are continuous and of exponential order, the corresponding Laplace transforms exist, and the Volterra integral operator of the second kind $J$ is invertable. Entering the functions (39) in Algorithm 1, we get the second order algebraic equation
$76 a 2 − 520 a + 475 = 0 ,$
which has the two real solutions
$a 1 ∗ = 130 − 15 35 38 , a 2 ∗ = 130 + 15 35 38 .$
For each of them the Algorithm 1 delivers the following corresponding solution of the given nonlinear CVFIE
$u 1 ( x ) = 35 − 15 152 x 3 + 6 x and u 2 ( x ) = − 35 + 15 152 x 3 + 6 x .$

#### 4.5. Example 5

Solve the nonlinear CVFIE of the second kind
$u ( x ) − ∫ 0 x e ( x − t ) u ( t ) d t − ∫ 0 1 1 u ( t ) d t = 1 , 0 ≤ x ≤ 1 .$
According to Theorem 2, we have $n = m = 1$, $b = 1$ and
$k 1 ( x ) = e x , g 1 ( x ) = 1 , h 1 ( t ) = 1 , φ ¯ 1 ( t , u ( t ) ) = 1 u ( t ) , f ( x ) = 1 .$
The functions $k 1 ( x )$, $g 1 ( x )$ and $f ( x )$ are continuous and of exponential order, and therefore the Volterra integral operator $J$ can be inverted. Entering the functions (41) in Algorithm 1, we get the algebraic equation
$a 2 + a + ln ( e 2 + 1 ) − ln 2 − 2 = 0 ,$
which admits the two real solutions
$a 1 ∗ = − − 4 ln ( e 2 + 1 ) + 4 ln 2 + 9 + 1 2 , a 2 ∗ = − 4 ln ( e 2 + 1 ) + 4 ln 2 + 9 − 1 2 .$
The corresponding solutions of the given nonlinear CVFIE (40) are
$u 1 ( x ) = − − 4 ln ( e 2 + 1 ) + 4 ln 2 + 9 − 1 4 e 2 x + 1 , u 2 ( x ) = − 4 ln ( e 2 + 1 ) + 4 ln 2 + 9 + 1 4 e 2 x + 1 .$

#### 4.6. Example 6

As a last example, we consider the nonlinear CVFIE of the second kind
$u ( x ) − ∫ 0 x sin ( x − t ) u ( t ) d t − ∫ 0 1 t e x − t ( t − u ( t ) ) 2 d t = sin x − e x , 0 ≤ x ≤ 1 ,$
The kernel $q ¯ ( x , t ) = e x − t = e x e − t$ is degenerate, and therefore Theorem 2 is applicable. We set $n = m = 1$, $b = 1$ and
$k 1 ( x ) = sin x , g 1 ( x ) = e x , h 1 ( t ) = e − t , φ ¯ 1 ( t , u ( t ) ) = t ( t − u ( t ) ) 2 , f ( x ) = sin x − e x .$
The functions $k 1 ( x )$, $g 1 ( x )$ and $f ( x )$ are continuous and of exponential order, and hence the inverse Volterra operator $J − 1$ exists. Entering (43) in Algorithm 1, we get the algebraic equation
$35 e − 84 a 2 + 168 − 73 e a + 35 e − 84 = 0 ,$
which has the two real solutions
$a 1 ∗ = − 3 e 143 e − 336 − 73 e + 168 70 e − 168 , a 2 ∗ = 3 e 143 e − 336 + 73 e − 168 70 e − 168 .$
Therefore, the given nonlinear CVFIE (42) has two real solutions that are
$u 1 ( x ) = − − 6 e x + 1 + 3 e 143 e − 336 2 e x − x − 1 + ( 168 − 67 e ) x + 3 e 70 e − 168 , u 2 ( x ) = 6 e x + 1 + 3 e 143 e − 336 2 e x − x − 1 + ( 67 e − 168 ) x − 3 e 70 e − 168 .$

## 5. Conclusions

Many different approximate and numerical methods are available in the literature for solving linear and nonlinear VFIE. However, there is no effective direct procedure for solving them in closed form.
In this paper, a direct operator method for the closed-form solution of some classes of VFIE was presented. An algorithm was developed for the case of VFIE with a linear Volterra convolution-type operator. The algorithm was implemented into the Maxima Computer Algebra System, and several problems were solved from the literature.
The main advantages of the technique are that it calculates the exact solution of VFIE and avoids problems related to approximate and numerical methods, such as computer accuracy errors and method convergence and stability. In the case of nonlinear VFIE, it finds all the solutions, rather than only one, as is the case with most numerical methods.
The only drawback of the method is that it requires the inverse of the Volterra operator and that all integrations must be performed analytically in general. In the special case of Volterra convolution-type operators, it depends upon the existence of the inverse Laplace transform of the involved functions. These limit the scope of the method to certain categories of functions, which makes perfect sense given the complexity of VFIE.
Solving some problems has proven that the method will be useful in several fields of science and engineering. The method can be extended to other categories of Volterra operators, the mixed type VFIE and the VFIE in two or three dimensions.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

The author would like to thank the anonymous reviewers for their valuable comments.

## Conflicts of Interest

The authors declare no conflict of interest.

## Abbreviations

The following abbreviations are used in this manuscript:
 IE Integral Equation(s) VIE Volterra Integral Equation(s) VFIE Volterra–Fredholm Integral Equation(s) CVIE Convolution-type Volterra Integral Equation(s) CVFIE Convolution-type Volterra-Fredholm Integral Equation(s)

## References

1. Tsokos, C.P.; Padgett, W.J. Random Integral Equations with Applications to Life Sciences and Engineering; Academic Press Inc.: London, UK, 1974; pp. ii–iv, ix–x, 1–278. [Google Scholar]
2. Prüss, J. Evolutionary Integral Equations and Applications; Springer: Basel, Switzerland, 1993. [Google Scholar]
3. Warnick, K.F. Numerical Analysis for Electromagnetic Integral Equations; Artech House: Norwood, MA, USA, 2008. [Google Scholar]
4. Amin, R.; Nazir, S.; García-Magariño, I. Efficient sustainable algorithm for numerical solution of nonlinear delay Fredholm-Volterra integral equations via Haar wavelet for dense sensor networks in emerging telecommunications. Trans. Emerg. Telecommun. Technol. 2020, 33, e3877. [Google Scholar] [CrossRef]
5. Wazwaz, A.M. Linear and Nonlinear Integral Equations; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar] [CrossRef]
6. Mamedov, Y.D.; Musaev, V.M. On the theory of the solutions of nonlinear operator equations. Dokl. Akad. Nauk SSSR 1970, 195, 36–39. [Google Scholar]
7. Ashirov, S.; Mamedov, Y.D. Investigation of solutions of nonlinear Volterra-Fredholm operator equations. Dokl. Akad. Nauk SSSR 1976, 229, 265–268. [Google Scholar]
8. Zaghrout, A.S.S. On Volterra-Fredholm integral equations. Period Math. Hung. 1993, 26, 55–64. [Google Scholar] [CrossRef]
9. Rao, V.S.H. On random solutions of Volterra-Fredholm integral equations. Pac. J. Math. 1983, 108, 397–405. [Google Scholar] [CrossRef]
10. Yalçinbaş, S. Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput. 2002, 127, 195–206. [Google Scholar] [CrossRef]
11. Balachandran, K.; Sumathy, K.; Kuo, H.H. Existence of Solutions of General Nonlinear Stochastic Volterra Fredholm Integral Equations. Stoch. Anal. Appl. 2005, 23, 827–851. [Google Scholar] [CrossRef]
12. Mahmoudi, Y. Taylor polynomial solution of non-linear Volterra-Fredholm integral equation. Int. J. Comput. Math. 2005, 82, 881–887. [Google Scholar] [CrossRef]
13. Yousefi, S.; Razzaghi, M. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Math. Comput. Simul. 2005, 70, 1–8. [Google Scholar] [CrossRef]
14. Cui, M.; Du, H. Representation of exact solution for the nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput. 2006, 182, 1795–1802. [Google Scholar] [CrossRef]
15. Li, X.F.; Fang, M. Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion. Int. J. Comput. Math. 2006, 83, 637–649. [Google Scholar] [CrossRef]
16. Babolian, E.; Fattahzadeh, F.; Golpar Raboky, E. A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type. Appl. Math. Comput. 2007, 189, 641–646. [Google Scholar] [CrossRef]
17. Bildik, N.; Inc, M. Modified decomposition method for nonlinear Volterra-Fredholm integral equations. Chaos Solitons Fract. 2007, 33, 308–313. [Google Scholar] [CrossRef]
18. Ghasemi, M.; Kajani, M.T.; Babolian, E. Numerical solutions of the nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method. Appl. Math. Comput. 2007, 188, 446–449. [Google Scholar] [CrossRef]
19. Ordokhani, Y.; Razzaghi, M. Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions. Appl. Math. Lett. 2008, 21, 4–9. [Google Scholar] [CrossRef]
20. Pachpatte, B.G. On a nonlinear Volterra-Fredholm integral equation. Sarajevo J. Math. 2008, 4, 61–71. [Google Scholar]
21. Caliò, F.; Fernández Muñoz, M.V.; Marchetti, E. Direct and iterative methods for the numerical solution of mixed integral equations. Appl. Math. Comput. 2010, 216, 3739–3746. [Google Scholar] [CrossRef]
22. Hendi, F.A.; Albugami, A.M. Numerical solution for Fredholm–Volterra integral equation of the second kind by using collocation and Galerkin methods. J. King Saud Univ. Sci. 2010, 22, 37–40. [Google Scholar] [CrossRef]
23. Maleknejad, K.; Almasieh, H.; Roodakiand, M. Triangular functions (TF) method for the solution of nonlinear Volterra-Fredholm integral equations. Commun. Nonlinear Sci. Numer. Simul. 2010, 15, 3293–3298. [Google Scholar] [CrossRef]
24. Fariborzi Araghi, M.A.; Behzadi, S.S. Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using Homotopy Analysis Method. J. Appl. Math. Comput. 2011, 37, 1–12. [Google Scholar] [CrossRef]
25. Attari, H.; Yazdani, A. A Computational Method for Fuzzy Volterra-Fredholm Integral Equations. Fuzzy Inf. Eng. 2011, 3, 147–156. [Google Scholar] [CrossRef]
26. Marzban, H.R.; Tabrizidooz, H.R.; Razzaghi, M. A composite collocation method for the nonlinear mixed Volterra—Fredholm Hammerstein integral equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16, 1186–1194. [Google Scholar] [CrossRef]
27. Shahsavaran, A. Numerical Solution of Nonlinear Fredholm-Volterra Integtral Equations via Piecewise Constant Function by Collocation Method. Am. J. Comput. Math. 2011, 1, 134–138. [Google Scholar] [CrossRef]
28. El-Ameen, M.A.; El-Kady, M. A New Direct Method for Solving Nonlinear Volterra-Fredholm-Hammerstein Integral Equations via Optimal Control Problem. J. Appl. Math. 2012, 2012, 714973. [Google Scholar] [CrossRef]
29. Caliò, F.; Garralda-Guillem, A.I.; Marchetti, E.; Ruiz Galán, M. About some numerical approaches for mixed integral equations. Appl. Math. Comput. 2012, 219, 464–474. [Google Scholar] [CrossRef]
30. Chen, Z.; Jiang, W. An approximate solution for a mixed linear Volterra-Fredholm integral equation. Appl. Math. Lett. 2012, 25, 1131–1134. [Google Scholar] [CrossRef]
31. Dastjerdi, H.L.; Ghaini, F.M.M. Numerical solution of Volterra-Fredholm integral equations by moving least square method and Chebyshev polynomials. Appl. Math. Model. 2012, 36, 3283–3288. [Google Scholar] [CrossRef]
32. Ezzati, R.; Mokhtari, F.; Maghasedi, M. Numerical Solution of Volterra-Fredholm Integral Equations with the Help of Inverse and Direct Discrete Fuzzy Transforms and Collocation Technique. Int. J. Ind. Math. 2012, 4, 221–229. [Google Scholar]
33. Gámez, D. Analysis of the Error in a Numerical Method Used to Solve Nonlinear Mixed Fredholm-Volterra-Hammerstein Integral Equations. J. Funct. Spaces Appl. 2012, 2012, 242870. [Google Scholar] [CrossRef]
34. Maleknejad, K.; Hashemizadeh, E.; Basirat, B. Computational method based on Bernstein operational matrices for nonlinear Volterra-Fredholm-Hammerstein integral equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 52–61. [Google Scholar] [CrossRef]
35. Parand, K.; Rad, J.A. Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions. Appl. Math. Comput. 2012, 218, 5292–5309. [Google Scholar] [CrossRef]
36. Shali, J.; Darania, P.; Akbarfam, A. Collocation Method for Nonlinear Volterra-Fredholm Integral Equations. Open J. Appl. Sci. 2012, 2, 115–121. [Google Scholar] [CrossRef]
37. Shamloo, A.; Shahkar, S.; Madadi, A. Numerical Solution of the Fredholme-Volterra Integral Equation by the Sinc Function. Am. J. Comput. Math. 2012, 2, 136–142. [Google Scholar] [CrossRef]
38. Hetmaniok, E.; Nowak, I.; Słota, D.; Wituła, R. A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra-Fredholm integral equations. Appl. Math. Lett. 2013, 26, 165–169. [Google Scholar] [CrossRef]
39. Al-Jarrah, Y.; Lin, E. Numerical Solution of Fredholm-Volterra Integral Equations by Using Scaling Function Interpolation Method. Appl. Math. 2013, 4, 204–209. [Google Scholar] [CrossRef]
40. Mesgarani, H.; Mollapourasl, R. Theoretical investigation on error analysis of Sinc approximation for mixed Volterra-Fredholm integral equation. Comput. Math. Math. Phys. 2013, 53, 530–539. [Google Scholar] [CrossRef]
41. Mirzaee, F.; Hoseini, A.A. “Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of…“ [Alexandria Eng. J. 52 (2013) 551–555]. Alex. Eng. J. 2019, 58, 1099–1102. [Google Scholar] [CrossRef]
42. Wang, K.; Wang, Q. Lagrange collocation method for solving Volterra-Fredholm integral equations. Appl. Math. Comput. 2013, 219, 10434–10440. [Google Scholar] [CrossRef]
43. Wang, K.; Wang, Q.; Guan, K. Iterative method and convergence analysis for a kind of mixed nonlinear Volterra-Fredholm integral equation. Appl. Math. Comput. 2013, 225, 631–637. [Google Scholar] [CrossRef]
44. Căruntu, B.; Bota, C. Polynomial least squares method for the solution of nonlinear Volterra-Fredholm integral equations. Math. Probl. Eng. 2014, 2014, 147079. [Google Scholar] [CrossRef]
45. Mashayekhi, S.; Razzaghi, M.; Tripak, O. Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials. Sci. World J. 2014, 2014, 413623. [Google Scholar] [CrossRef]
46. Wang, X. A New Wavelet Method for Solving a Class of Nonlinear Volterra-Fredholm Integral Equations. Abstr. Appl. Anal. 2014, 2014, 975985. [Google Scholar] [CrossRef]
47. Wang, K.; Wang, Q. Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations. J. Comput. Appl. Math. 2014, 260, 294–300. [Google Scholar] [CrossRef]
48. Wang, Q.; Wang, K.; Chen, S. Least squares approximation method for the solution of Volterra-Fredholm integral equations. J. Comput. Appl. Math. 2014, 272, 141–147. [Google Scholar] [CrossRef]
49. Zulkarnain, F.; Eshkuvatov, Z.; Muminov, Z.; Nik Long, N. Modified Decomposition Method for Solving Nonlinear Volterra-Fredholm Integral Equations. In Proceedings of the International Conference on Mathematical Sciences and Statistics, Kuala Lumpur, Malaysia, 5–7 February 2013; Kilicman, A., Leong, W., Eshkuvatov, Z., Eds.; Springer: Singapore, 2014; pp. 103–110. [Google Scholar] [CrossRef]
50. Hosseini, S.A.; Shahmorad, S.; Talati, F. A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations. Numer. Algor. 2015, 68, 511–529. [Google Scholar] [CrossRef]
51. Micula, S. An iterative numerical method for Fredholm–Volterra integral equations of the second kind. Appl. Math. Comput. 2015, 270, 935–942. [Google Scholar] [CrossRef]
52. Mirzaee, F.; Hadadiyan, E. Applying the modified block-pulse functions to solve the three-dimensional Volterra-Fredholm integral equations. Appl. Math. Comput. 2015, 265, 759–767. [Google Scholar] [CrossRef]
53. Das, P.; Nelakanti, G. Convergence Analysis of Legendre Spectral Galerkin Method for Volterra-Fredholm-Hammerstein Integral Equations. In Mathematical Analysis and its Applications; Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H., Eds.; Springer: New Delhi, India, 2015; pp. 3–15. [Google Scholar] [CrossRef]
54. Tang, X. Numerical solution of Volterra-Fredholm integral equations using parameterized pseudospectral integration matrices. Appl. Math. Comput. 2015, 270, 744–755. [Google Scholar] [CrossRef]
55. Gouyandeh, Z.; Allahviranloo, T.; Armand, A. Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via Tau-collocation method with convergence analysis. J. Comput. Appl. Math. 2016, 308, 435–446. [Google Scholar] [CrossRef]
56. Mirzaee, F.; Hadadiyan, E. Numerical solution of Volterra-Fredholm integral equations via modification of hat functions. Appl. Math. Comput. 2016, 280, 110–123. [Google Scholar] [CrossRef]
57. Mirzaee, F.; Hoseini, S.F. Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations. Appl. Math. Comput. 2016, 273, 637–644. [Google Scholar] [CrossRef]
58. Berenguer, M.I.; Gámez, D. Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm-Volterra integral equations of Hammerstein type. Appl. Anal. 2017, 96, 516–527. [Google Scholar] [CrossRef]
59. Dahaghin, M.S.; Eskandari, S. Solving two-dimensional Volterra-Fredholm integral equations of the second kind by using Bernstein polynomials. Appl. Math. J. Chin. Univ. 2017, 32, 68–78. [Google Scholar] [CrossRef]
60. Davaeifar, S.; Rashidinia, J. Boubaker polynomials collocation approach for solving systems of nonlinear Volterra-Fredholm integral equations. J. Taibah Univ. Sci. 2017, 11, 1182–1199. [Google Scholar] [CrossRef]
61. Fallahpour, M.; Khodabin, M.; Maleknejad, K. Theoretical error analysis and validation in numerical solution of two-dimensional linear stochastic Volterra-Fredholm integral equation by applying the block-pulse functions. Cogent Math. 2017, 4, 1296750. [Google Scholar] [CrossRef]
62. Hesameddini, E.; Shahbazi, M. Solving system of Volterra-Fredholm integral equations with Bernstein polynomials and hybrid Bernstein Block-Pulse functions. J. Comput. Appl. Math. 2017, 315, 182–194. [Google Scholar] [CrossRef]
63. Hesameddini, E.; Khorramizadeh, M.; Shahbazi, M. Bernstein polynomials method for solving Volterra-Fredholm integral equations. Bull. Mathématique Société Des Sci. Mathématiques Roum. 2017, 60, 59–68. [Google Scholar]
64. Ma, Y.; Huang, J.; Wang, C. Numerical Solutions of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations Using Sinc Nyström Method. In Information Technology and Intelligent Transportation Systems; Balas, V., Jain, L., Zhao, X., Eds.; Advances in Intelligent Systems and Computing; Springer: Cham, Switzerland, 2017; Volume 455, pp. 187–194. [Google Scholar] [CrossRef]
65. Mirzaee, F. Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials. Comput. Methods Differ. Equ. 2017, 5, 88–102. [Google Scholar]
66. Shiralashetti, S.C.; Mundewadi, R.A. Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method. Bull. Math. Sci. Appl. 2017, 18, 51. [Google Scholar] [CrossRef]
67. Xie, J.; Huang, Q.; Zhao, F. Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces based on Block Pulse functions. J. Comput. Appl. Math. 2017, 317, 565–572. [Google Scholar] [CrossRef]
68. Erfanian, M. The approximate solution of nonlinear mixed Volterra-Fredholm-Hammerstein integral equations with RH wavelet bases in a complex plane. Math. Methods Appl. Sci. 2018, 41, 8942–8952. [Google Scholar] [CrossRef]
69. Erfanian, M.; Zeidabadi, H. Using of Bernstein spectral Galerkin method for solving of weakly singular Volterra-Fredholm integral equations. Math. Sci. 2018, 12, 103–109. [Google Scholar] [CrossRef]
70. Negarchi, N.; Nouri, K. Numerical solution of Volterra-Fredholm integral equations using the collocation method based on a special form of the Müntz–Legendre polynomials. J. Comput. Appl. Math. 2018, 344, 15–24. [Google Scholar] [CrossRef]
71. Abdeljawad, T.; Agarwal, R.P.; Karapınar, E.; Kumari, P.S. Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space. Symmetry 2019, 11, 686. [Google Scholar] [CrossRef]
72. Van Ninh, K.; Binh, N.T. Analytical Solution of Volterra-Fredholm Integral Equations Using Hybrid of the Method of Contractive Mapping and Parameter Continuation Method. Int. J. Appl. Comput. Math 2019, 5, 76. [Google Scholar] [CrossRef]
73. Sabzevari, M. A review on “Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of …” [Alexandria Eng. J. 52 (2013) 551–555]. Alex. Eng. J. 2019, 58, 1099–1102. [Google Scholar] [CrossRef]
74. Tomasiello, S.; Macías-Díaz, J.E.; Khastan, A.; Alijani, Z. New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra-Fredholm integral equations. Neural Comput. Applic. 2019, 31, 4865–4878. [Google Scholar] [CrossRef]
75. Amiri, S.; Hajipour, M.; Baleanu, D. A spectral collocation method with piecewise trigonometric basis functions for nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput. 2020, 370, 124915. [Google Scholar] [CrossRef]
76. Al-Bugami, A.; Al-Juaid, J. Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel. J. Appl. Math. Phys. 2020, 8, 2043–2054. [Google Scholar] [CrossRef]
77. Deniz, S. Optimal perturbation iteration technique for solving nonlinear Volterra-Fredholm integral equations. Math. Methods Appl. Sci. 2020, 1–7. [Google Scholar] [CrossRef]
78. Du, H.; Chen, Z. A new reproducing kernel method with higher convergence order for solving a Volterra-Fredholm integral equation. Appl. Math. Lett. 2020, 102, 106117. [Google Scholar] [CrossRef]
79. Georgieva, A.; Hristova, S. Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations. Fractal Fract. 2020, 4, 9. [Google Scholar] [CrossRef]
80. Bin Jebreen, H. On the Multiwavelets Galerkin Solution of the Volterra-Fredholm Integral Equations by an Efficient Algorithm. J. Math. 2020, 2020, 2672683. [Google Scholar] [CrossRef]
81. Youssri, Y.H.; Hafez, R.M. Chebyshev collocation treatment of Volterra-Fredholm integral equation with error analysis. Arab. J. Math. 2020, 9, 471–480. [Google Scholar] [CrossRef]
82. Abdou, M.A.; Elhamaky, M.N.; Soliman, A.A.; Mosa, G.A. The behaviour of the maximum and minimum error for Fredholm-Volterra integral equations in two-dimensional space. J. Interdiscip. Math. 2021, 24, 2049–2070. [Google Scholar] [CrossRef]
83. Dobriţoiu, M. The Existence and Uniqueness of the Solution of a Nonlinear Fredholm–Volterra Integral Equation with Modified Argument via Geraghty Contractions. Mathematics 2021, 9, 29. [Google Scholar] [CrossRef]
84. Esmaeili, H.; Rostami, M.; Hooshyarbakhsh, V. Numerical solution of Volterra-Fredholm integral equation via hyperbolic basis functions. Int. J. Numer. Model. 2021, 34, e2823. [Google Scholar] [CrossRef]
85. Geçmen, M.Z.; Çelik, E. Numerical solution of Volterra-Fredholm integral equations with Hosoya polynomials. Math. Methods Appl. Sci. 2021, 44, 11166–11173. [Google Scholar] [CrossRef]
86. He, J.-H.; Taha, M.H.; Ramadan, M.A.; Moatimid, G.M. Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations. Axioms 2021, 10, 200. [Google Scholar] [CrossRef]
87. Mohamed, A.S. Spectral Solutions with Error Analysis of Volterra-Fredholm Integral Equation via Generalized Lucas Collocation Method. Int. J. Appl. Comput. Math 2021, 7, 178. [Google Scholar] [CrossRef]
88. Xu, M.; Niu, J.; Tohidi, E.; Hou, J.; Jiang, D. A new least-squares-based reproducing kernel method for solving regular and weakly singular Volterra-Fredholm integral equations with smooth and nonsmooth solutions. Math. Methods Appl. Sci. 2021, 44, 10772–10784. [Google Scholar] [CrossRef]
89. Hamani, F.; Rahmoune, A. Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method. Tatra Mt. Math. Publ. 2021, 80, 35–52. [Google Scholar] [CrossRef]
90. Ramadan, M.A.; Osheba, H.S.; Hadhoud, A.R. A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra-Fredholm integral equations. Numer. Methods Partial. Differ. Equ. 2022, 1–13. [Google Scholar] [CrossRef]
91. Pachpatte, B.G. On mixed Volterra-Fredholm type integral equations. Indian J. Pure Appl. Math. 1986, 17, 488–496. [Google Scholar]
92. Kauthen, J.P. Continuous time collocation methods for Volterra-Fredholm integral equations. Numer. Math. 1989, 56, 409–424. [Google Scholar] [CrossRef]
93. Brunner, H. On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations by Collocation Methods. SIAM J. Numer. Anal. 1990, 27, 987–1000. [Google Scholar] [CrossRef]
94. Rizk, M.M.; Zaher, S.L. On the approximate solution of nonlinear Volterra-Fredholm integral equations on a complex domain by Dzyadyk’s method. Ukr. Math. J. 1997, 49, 1705–1717. [Google Scholar] [CrossRef]
95. Maleknejad, K.; Hadizadeh, M. A new computational method for Volterra-Fredholm integral equations. Comput. Math. Appl. 1999, 37, 1–8. [Google Scholar] [CrossRef]
96. Wazwaz, A.M. A reliable treatment for mixed Volterra-Fredholm integral equations. Appl. Math. Comput. 2002, 127, 405–414. [Google Scholar] [CrossRef]
97. Banifatemi, E.; Razzaghi, M.; Yousefi, S. Two-dimensional Legendre wavelets method for the mixed Volterra-Fredholm integral equations. J. Vib. Control 2007, 13, 1667–1675. [Google Scholar] [CrossRef]
98. Berenguer, M.I.; Gámez, D.; López Linares, A.J. Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm–Volterra integro-differential equation. J. Comput. Appl. Math. 2013, 252, 52–61. [Google Scholar] [CrossRef]
99. Dong, C.; Chen, Z.; Jiang, W. A modified homotopy perturbation method for solving the nonlinear mixed Volterra-Fredholm integral equation. J. Comput. Appl. Math. 2013, 239, 359–366. [Google Scholar] [CrossRef]
100. Mirzaee, F.; Hadadiyan, E. Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra-Fredholm integral equations. Alex. Eng. J. 2016, 55, 2943–2952. [Google Scholar] [CrossRef]
101. Micula, S. On Some Iterative Numerical Methods for Mixed Volterra-Fredholm Integral Equations. Symmetry 2019, 11, 1200. [Google Scholar] [CrossRef]
102. Micula, S. Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind. Symmetry 2021, 13, 1326. [Google Scholar] [CrossRef]
103. El Majouti, Z.; El Jid, R.; Hajjaj, A. Numerical solution for three-dimensional nonlinear mixed Volterra-Fredholm integral equations via modified moving least-square method. Int. J. Comput. Math. 2021. [Google Scholar] [CrossRef]
104. Polyanin, A.D.; Manzhirov, A.V. Handbook of Integral Equations, 2nd ed.; Chapman and Hall/CRC: New York, NY, USA, 2008. [Google Scholar] [CrossRef]
105. Parasidis, I.N.; Providas, E. Extension Operator Method for the Exact Solution of Integro-Differential Equations. In Contributions in Mathematics and Engineering: In Honor of Constantin Carathéodory; Pardalos, P.M., Rassias, T.M., Eds.; Springer International Publishing: Cham, Switzerland, 2016; pp. 473–496. [Google Scholar] [CrossRef]
106. Parasidis, I.N.; Providas, E. On the Exact Solution of Nonlinear Integro-Differential Equations. In Applications of Nonlinear Analysis; Rassias, T., Ed.; Springer: Cham, Switzerland, 2018; pp. 591–609. [Google Scholar] [CrossRef]
107. Baiburin, M.M.; Providas, E. Exact Solution to Systems of Linear First-Order Integro-Differential Equations with Multipoint and Integral Conditions. In Mathematical Analysis and Applications; Rassias, T.M., Pardalos, P.M., Eds.; Springer: Cham, Switzerland, 2019; pp. 1–16. [Google Scholar] [CrossRef]
108. Providas, E. Approximate Solution of Fredholm Integral and Integro-Differential Equations with Non-Separable Kernels. In Approximation and Computation in Science and Engineering; Daras, N.J., Rassias, T.M., Eds.; Springer: Cham, Switzerland, 2022; pp. 693–708. [Google Scholar] [CrossRef]
109. Kreyszig, E. Advanced Engineering Mathematics, 7th ed.; John Wiley & Sons.: Singapore, 1993. [Google Scholar]
110. Rezaei, H.; Jung, S.M.; Rassias, T.M. Laplace transform and Hyers–Ulam stability of linear differential equations. J. Math. Anal. Appl. 2013, 403, 244–251. [Google Scholar] [CrossRef]
111. Kagiwada, H.H.; Kalaba, R.E. An initial value method for fredholm integral equations of convolution type. Int. J. Comput. Math. 1968, 2, 143–155. [Google Scholar] [CrossRef]
112. Tuna, M.; Kirca, M. Exact solution of Eringen’s nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams. Int. J. Eng. Sci. 2016, 105, 80–92. [Google Scholar] [CrossRef]
113. Providas, E. On the exact solution of nonlocal Euler-Bernoulli beam equations via a direct approach for Volterra-Fredholm integro-differential equations. Appliedmath, 2022; under review. [Google Scholar]
114. Providas, E. Closed-form solution of the bending two-phase integral model of Euler-Bernoulli nanobeams. Algorithms 2022, 15, 151. [Google Scholar] [CrossRef]
Table 1. Exact solution of the linear CVFIE (33) for three different cases of the input function $f ( x )$.
Table 1. Exact solution of the linear CVFIE (33) for three different cases of the input function $f ( x )$.
$f ( x )$$u ( x )$
$α x 2 + β x + γ , α , β , γ ∈ R$$− 3 α x 4 + 6 β x 3 + ( 18 γ − 36 α ) x 2 − ( 18 γ + 44 β + 5 α ) x − 54 γ − 8 β − 5 α 36$
$sin ( π k x ) , k ∈ Z +$$( 3 π 2 k 2 + 3 ) sin ( π k x ) − [ sin ( π k ) + π k cos ( π k ) + π k ] x − sin ( π k ) − π k cos ( π k ) + 2 π k 3 π 2 k 2$
$x 3 cosh ( x )$$( 18 x 2 − 54 x + 72 ) e x − ( 18 x 2 + 54 x + 72 ) e − x − ( 13 e + 12 − 33 e − 1 ) x − 13 e + 24 + 33 e − 1 6$
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Providas, E. An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type. Algorithms 2022, 15, 203. https://doi.org/10.3390/a15060203

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Providas E. An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type. Algorithms. 2022; 15(6):203. https://doi.org/10.3390/a15060203

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Providas, Efthimios. 2022. "An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type" Algorithms 15, no. 6: 203. https://doi.org/10.3390/a15060203

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