In this study, we consider the following linear Volterra integro-diferential equations (or only VIDEs) with initial conditions. The linear VIDEs of first kind are given by

where and are suficiently smooth functions. The linear VIDEs of second kind are expressed by

where and are suficiently smooth functions. The functions and are called kernels of the linear VIDEs.

Let be defined and piecewise continuous function for all positive values of and be of exponential order. The Laplacetransform of is defined by

where is said to be the inverseLaplace transform of , denoted by . Let us recall some useful results on the Laplace transform that shall be used in the next hereinafter.

(1) The Laplace transform of some functions:

(2) If LⁿF (x)^o = G(s) then

(3) If LⁿF (x)^o = G(s) then · · · − F ⁽ⁿ⁻¹⁾(0).

(4) If then

(5) The convolution of two functions and , denoted by is defined by . Let and . Then the convolution theorem says that

(6) The inverse Laplace transform of some functions:

The Laguerre polynomial is a polynomial function given by

where and are called the degree and the index of the Laguerre polynomial, respectively. Some important results on the Laguerre polynomials that shall be referred in the next as follows:

The Volterra integro-diferential equations are typically mathematical models in many areas of science and engineering. Solutions of these equations play vital roles in a number of processes and phenomena such as nuclear reactors, circuit analyses, wave propagation, glass forming processes, nano-hydrodynamics, visco elasticity, biological populations, etc. Therefore, there are many researchers who have been interested in the VIDEs and founded numerous methods to solve the analytical and numerical solutions of VIDEs up to the present as follows. Estimated solutions of nonlinear VIDEs of a fractional order were investigated applying the Laplace transform and the Adomian polynomials by Yang and Hou [1]. Moreover, the Legendre polynomial approximation was used to find numerical solutions of nonlinear VIDEs of second kind by Gachpazan, Erfanian and Beiglo [2]. In addition, analytical solutions of linear VIDEs of second kind were solved using the Kamal transform by Aggarwal and Gupta [3]. The modified Adomian decomposition method was utilized to explain exact solutions of linear VIDEs of second kind by Okai, Ilejimi and Ibrahim [4]. Furthermore, approximate solutions of nonlinear VIDEs involving delay were found taking a new higher order method by Jhinga, Patade and Gejji [5]. Other than those findings, the Sadik transform was applied to figure out exact solutions of first kind VIDEs on convolution type kernels by Aggarwal, Vyas and Sharma [6]. Numerical solutions of linear VIDEs were estimated using the Laguerre and Touchard polynomials by Abdullah and Ali [7]. So far, some asymptotic behavior of exact solutions of nonlinear VIDEs has been studied by Cakir, Gunes and Duru [8]. The quasilinearization technique to diference scheme also has been applied to solve estimated solutions of VIDEs in [8]. Recently, the asymptotic behavior of the analytical solutions of the singularly perturbed nonlinear VIDEs has been established by Cakir, Cakir and Cakir [9]. The uniform diference scheme on the Bakhvalov-Shishkin mesh points according to the boundary layer conditions has been introduced to find numerical solutions of VIDEs as well in [9]. Exact solutions of the Faltung type VIDEs for first kind have been solved applying the Kushare transform by Patil, Nikam and Shinde [10].

In this research, we compound linear Volterra integro-diferential equations of first and second kinds to be a general form. Then, we take the Laplace transform to find an exact solution on a convolution type and utilize the Laguerre polynomials to estimate a solution on a non-convolution type of this generalization.

For this section, we assume that and start to introduce a generalization of linear Volterra integro-diferential equations expressed by

where is an exponentially bounded and smooth function, and are exponentially bounded and suficiently smooth functions and are real numbers. and or and , then the equation (3) is the linear VIDE of first kind and we can see [6] and [10] for more vital results. If and , then the equation (3) is the linear VIDE of second kind and we can see [1], [2], [3], [4], [5], [7], [8] and [9] for more comprehensive findings.

Now, we will focus on a solution of this generalization on a convolution type with a constant function A, that is, we then consider the following initial-value problem as follows:

We will utilize the Laplace transform to solve the problem as the following steps: At the beginning, applying the Laplace transform to (4), we get

After, using the convolution theorem to (5), we then obtain

Then, taking the Laplace transform of derivatives on (6) with initial conditions, we have

and we also obtain

At last, operating the inverse Laplace transform on (7), we receive the solution of initial-value problem (4) as follows.

Example 2.1Solve the Volterra integro-differential problem:

Solution. Firstly, applying Laplace transform to the problem, we have

Secondly, using the convolution theorem, we get

Thirdly, taking the Laplace transform of derivatives, we have

and using initial conditions, we also obtain

Fourthly, rearranging the equation, we certainly receive

or

Thus, we get

Finally, taking the inverse Laplace transform of the equation, we suddenly have an analytical solution

Example 2.2.Solve the Volterra integro-diferential problem:

Solution. The first one, applying Laplace transform to the problem, we have

The second one, using the convolution theorem, we get

The third one, taking the Laplace transform of derivatives, we have

and using initial conditions, we also obtain

The fourth one, rearranging the equation, we certainly receive

or

Hence, we get

The last one, taking the inverse Laplace transform of the equation, we suddenly have an exact solution

Here, we will emphasize on an approximate solution of this generalization for a non-convolution type given by

The approximation using the Laguerre polynomials is below: To start with Supposing that the function is an estimated solution of the equation (8) defined by

where are the Laguerre polynomials and are unknown constants, . Then, writing equation (9) as a dot product, we have

Next, rearranging the equation (10) in a matrix formula, we also have

where are known constants. After that, finding the derivatives of , we have as follows:

and

Then, substituting the equation (11) into the equation (8), we receive

Simplifying and integrating the equation (12), we then have the new equation with unknown constants . In order to determine , using n initial conditions and selecting , with substituting in the new equation, we get a system of linear algrbraic equations of unknown constants. Solving this system by a program, we have the values of the unknown constants, that is, the numerical solution of the initial-value problem (8) is obtained.

In order to guarantee the convergence of this method, we will verify as follows. Let be an analytical solution of initial-value problem (8) that has derivatives of all orders at . Then, the Taylor series of is defined by

Thus, by the definition and process to find , we obtain that

Here, it is suficient to show that converges to 0 as to confirm that converges to 0 as . Since , we get . It is easy to determine that and converge to 0 as as . This means that converges to 0 as to confirm the convergence.

Example 2.3.Estimate a solution of the linear Volterra integro-diferential problem using . An exact solution is

Solution. First, suppose that a function is an approximate solution of this problem, that is,

Second, finding derivatives of , we have as follows:

and

Third, substituting the derivatives into the problem, we receive

Fourth, simplifying and integrating the equation, we obtain the new equation. Selecting and to substitute in the new equation with using 5 initial conditions, we get the following system:

At last, solving the system by a program, we have

Therefore, the numerical solution is

Example 2.4.Approximate a solution of the linear Volterra integro-diferential problem usingand 4x cos . An exact solution is

Solution. Firstly, let is an approximate solution of this problem, that is,

Secondly, finding derivatives of , we have as follows:

and

Thirdly, substituting the derivatives into the problem, we receive

Fourthly, simplifying and integrating the equation, we obtain the new equation. Selecting and to substitute in the new equation with using 3 initial conditions, we get the following system:

Finally, solving the system by a program, we get

Therefore, the numerical solution is

For the approximate solution , the first one, let be an approximate solution of this problem, that is,

The second one, finding derivatives of , we have as follows:

and

The third one, substituting the derivatives into the problem, we receive

The fourth one, simplifying and integrating the equation, we obtain the new equation. Selecting and to substitute in the new equation with using 3 initial conditions, we get the following system:

The last one, solving the system by a program, we get

Therefore, the numerical solution is

To demonstrate the practical applicability of the proposed method, we consider a classical population model with memory efects, where the current growth rate depends not only on the present population but also on its past values. Let denote the population size at time . The population growth with memory is defined by

where is the intrinsic growth rate and is a memory kernel. For the growth model of Drosophila, we get and the exponentially decaying kernel and see more in [11]. Thus, the growth model is in the form

This kernel is smooth, exponentially bounded and therefore satisfies the Laplacetransform conditions and other regularity assumptions used in our analysis. Laplace transform, the exact solution of the problem is Then, applying the proposed Laguerre method with and on we obatian the results as follows.

and

In this paper, a generalization of linear VIDEs has been introduced already. In general, all results show that the Laplace transform has been efective to solve analytical solutions of the generalization on convolution type kernels repeatedly and the Laguerre polynomials have been successful to figure out numerical solutions of the generalization on non-convolution type kernels several times. However, the Kushare transform, Sadik transform and Kamal transform are other methods that can be analytically solved on convolution types of this generalization similarly. Moreover, the main advantage of this analytical method is the fact that it gives the exact solutions in just few processes and uses very less computational work. We also suggest that this numerical method can be applicable to singularly perturbed linear VIDEs, which are one case of this generalization, to obtain accurately approximate solutions.