## 1 Introduction  
Volterra-Fredholm integro-differential equations (VFIDEs) are encountered in many different branches of science and engineering [1, 2, 3, 4]. As they are extremely difficult and in some cases even impossible to solve in exact closed form, many different techniques have been developed to construct approximate and numerical solutions. Some of these methods are mentioned below. Two well-known, traditional techniques are the series solution method [1, 5] and the variational iteration method [1]. A powerful and widely used method for solving linear and nonlinear integro-differential equations including VFIDEs is the Adomian decomposition method (ADM) and its variant, known as modified ADM [6, 7]. Methods employing Taylor polynomial expansions were developed and applied to find the approximate solution of high-order linear VFIDEs under mixed conditions [8] and high-order nonlinear VFIDEs [9, 10]. Collocation methods based on Haar wavelets and Legendre wavelets for the numerical solution of nonlinear VFIDEs were presented in [3, 11], respectively. An approach using hybrid Legendre polynomials and block-pulse functions was also reported in [12]. The authors of [13] applied the polynomial least squares method to construct approximate analytical solutions for a very general class of nonlinear VFIDEs. A moving least squares (MLS) method was developed in [14]. The Tau method was also used for the numerical solution of the general form of linear VFIDEs [15]. In [16], the reproducing kernel Hilbert space method was employed to numerically solve first-order periodic VFIDEs. A method based on triangular functions and the relative operational matrix to determine the numerical solution of specific nonlinear VFIDEs was proposed in [17]. Fixed-point techniques for the approximate solution of linear and first-order nonlinear VFIDEs were discussed in [18; 19], respectively. A quadrature and iterative technique for solving a nonlinear VFIDE with weakly singular kernels was developed in [20]. Numerical techniques for solving fractional VFIDEs were proposed in [21; 22; 23] using second Chebyshev wavelets (SCWs), fractional-order Bernoulli functions (FBFs) and Lucas wavelets (LWs), respectively. The numerical solution of multipoint boundary value problems for linear VFIDEs was studied using Legendre polynomials [24; 25], and Bernstein polynomials [26; 27]. A semi-analytical numerical method for solving multipoint boundary value problems for linear VFIDEs of the neutral type with linear functional arguments was presented in [28]. Finally, for other related types of problems, such as ordinary differential equations under multipoint and integral conditions and nonlinear Volterra equations with loads, see [29; 30]. This paper deals with the exact closed-form solution of a class of Volterra-Fredholm integro-differential equations of the Hammerstein type. It is emphasized that due to the complexity involved in solving integro-differential equations (IDEs) and in contrast to numerical methods, exact analytical techniques cannot have a general purpose but are designed and applied to a specific class of IDEs. Methods for solving explicitly different types of Volterra and Fredholm IDEs are discussed in detail in [31]. A method for the exact solution of systems of first-order linear Fredholm IDEs under multipoint and integral conditions was presented in [32]. The exact solution of linear and nonlinear convolution-type Volterra-Fredholm integral equations (VFIEs) was efficiently treated in [33].