Numerical Solutions of Mixed Integro-Differential Equations by Least-Squares Method and Laguerre Polynomial

The main objective of this article is to present a new technique for solving integro-differential equations (IDEs) subject to mixed conditions, based on the least-squares method (LSM) and Laguerre polynomial. To explain the effect of the proposed procedure will be discussed three examples of the first, second and three-order linear mixed IDEs. The numerical results used to demonstrate the validity and applicability of comparisons of this method with the exact solution shown that the competence and accuracy of the present method.


Introduction
Studied a sixth-kind Chebyshev collocation scheme will be considered for solving a category of (V-OFNQIDEs). The operational matrix of variable-order fractional derivative for sixth kind Chebyshev polynomials is derived and then, a clustering approach is employed to reduce the V-OFNQIDE to a system of nonlinear algebraic equations [1]. Developed a numerical method was implemented for solving IDEs with the weakly singular by using a new method depend on the cubic B-spline least-square method and a quadratic B-spline as a weight function. The numerical results are in suitable agreement with the exact solutions via calculating and norms errors. Theoretically, discussed the stability evaluation of the current method using the Von-Neumann method, which explained that this technique is unconditionally stable [2]. A new method based on the Laplace Adomian decomposition with the Bernstein 310 polynomial to solve the Voltaire integral equation and the integro-differential equation of the first and second types, and through examples and comparison accurate and approximate solutions the method is adopted [3]. Used waves and orthogonal polynomials to solve some of the system single Volterra integro-differential equations (SSVIDEs) presented the convergence analysis for the derivative of the approximation in terms of Legendre wavelets, to demonstrate the accuracy and efficiency of the proposed method through the given examples [4]. Developed is a new numerical scheme based on the Laguerre and Taylor polynomials, called matrix collocation scheme, solution nonlinear PIDEs, for the numerical solution of the mentioned nonlinear equations under the initial or boundary conditions [5]. The Euler polynomial was used to solve the (VIDEs) of the pantograph delay type in approximation of the solution. The method is discussed in detail and compared through numerical examples [6].
In this paper, we will be applying the least-squares method and Laguerre polynomial for solving linear mixed IDEs of the second type. The mixed IDEs in this method are transformed into a system of linear algebraic equations. Consider the following general forms of the mixed IDEs [7]:

Least-Square Method and Laguerre Polynomial
In this section, we will be implementing a new approach based on the LSM and Laguerre polynomial as a basis function to solve the equations (1.1) and (1.2).
We assume the approximate solution as
It is possible to describe another type of (2.12) by applying the conditions as. The solution of (1.1) under conditions (1.2) can be get by changing the row matrices (2.13) by the last (A) rows of the matrix form (2.12) we get the new augmented matrix [12,13,14].
therefore, the matrix A is uniquely determined. Also, the equation (1.1) with conditions (1.2) has a unique solution [1].

Convergence Analysis
Based on the numerical methods introduced in the second section, we will review an approximation of the above errors to prove that the approximate solution / ( ) will converge to the exact solution ( ) of (1.1).
Lemma [15]. Let ( ) ∈ j (ℝ l ), such that is the best approximation polynomial of ( ) for any A ≥ 0, and 0 ≤ ≥ A, we have where C is a positive constant, which depends on A.
Proof. Let m / denote the symmetric truncation to the grade, i.e., we get

Illustrative Examples
In this section, we provide numerical experiments of the suggested method. In all examples, set the parameter A = 5 and denote the exact solution and approximate solution by ( ) and / ( ) respectively.The error estimation is given to show the accuracy of the proposed method, and the following absolute error between the exact and approximate solutions is given by: In Figures 1-3, we can observe that the approximate solution is highly consistent with the exact solution for different values of x. Furthermore, in Tables 1-3, the absolute errors and at A = 5 are reported, which shows the improved rate of convergence comparing with other mentioned methods [16,17,18]. Example 2 [17]. Consider the following linear mixed IDE second kind Solution. Using the combination of least squares with the Laguerre polynomial defined in the form, an approximate solution of ( ) as (2.1) will be applied, when A = 5. Example 3 [18]. Considered the following linear mixed IDE second kind.

Conclusions
In this paper, we developed a method based on the least-squares method (LSM) and Laguerre polynomials to solve mixed integro-differential equations. To illustrate the effectiveness of the proposed method, three examples were solved based on the present technique. In addition, the convergence of this method was analyzed. The results show that the proposed method is practically reliable and consistent in comparison with other mentioned methods [16,17,18].