## 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, 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  
\[y(x)=f(x)+\int_{a}^{x}k_{1}(x,t)y(t)dt+\int_{a}^{b}k_{2}(x,t)y(t)dt,\quad a \leq x\leq b,\]  
and  
\[y(x)=f(x)+\int_{a}^{x}\int_{a}^{b}k(s,t)y(t)dtds,\quad a\leq x\leq b,\]  
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], [PERSON] polynomials [65], first Boubaker polynomials [60], Muntz-Legendre polynomials [70], generalized [PERSON] 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 [PERSON]'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], [PERSON]'s iterative approximation method [94] and [PERSON] 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.