Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce.

In recent years, many different basic functions have been used to estimate the solution of integral equations, such as orthogonal functions and wavelets. Three families of the orthogonal functions are classified: piecewise constant orthogonal functions (e.g., Walsh, Haar, block-pulse, etc.), orthogonal polynomials (e.g., Legendre, Laguerre, Chebyshev, etc.) and sine-cosine functions in the Fourier series. For instance, many authors investigated the general k th order integro-differential equation with initial conditions y(a) = a 0 , . . . , y (n−1) (a) = a n−1 , where a 0 , . . . , a n−1 are real constants, k, m are positive integers and m < k, the functions l, f, g are given and y(t) is the solution to be determined. In [9], the authors applied the homotopy perturbation method to solve Equation (1), while in [1,27], the authors changed the equation to an ordinary integro-differential equation and applied the variational iteration method to solve it so that the Lagrange multipliers can be effectively identified. Using the operational matrix of derivatives of hybrid functions, a numerical method has been presented in [13] to solve Equation (1). In [12], Hemeda used the iterative method introduced in [5] to solve the more general equation where n ≤ m < q.
In this paper, we use block-pulse and hybrid functions to approximate solutions y(t) of the Fredholm integro-differential system given by and solutions y(t) of the Volterra integro-differential system given by Here m, n are positive integers, l = max(m, n)−1, a 0 , . . . , a l are initial conditions, the parameters β, λ and the functions k(t, s), g(t, s) and f (t) are known and belong to L 2 [0, 1). The function y(t) as well as its derivatives y (n) and y (m) are unknown. We point out that System (3) is a particular case of Equation (2).
Hybrid functions have been applied extensively for solving differential systems and proved to be a useful mathematical tool. The pioneering work via hybrid functions was led by the authors in [18,25], who first derived an operational matrix for the integral of the hybrid function vector, and paved the way for the hybrid function analysis of the dynamic systems. Since then, the hybrid functions' approach has been improved and used to approximate differential equations or systems (see [15,17,21,22,19,23,20] and the references therein).
The novelty and the key point in solving Systems (3) and (4) are to use some useful properties of hybrid functions to derive a new approximation Y (n) of the derivative y (n) (t) of order n of the solution y(t) (see Lemma 3.1). Hence, Systems (3) and (4) can be converted into reduced algebraic systems.
For arbitrary positive integers q and r, the set {b km (t)}, k = 1, 2, . . . , q, m = 0, 1, . . . , r − 1 of hybrid functions will be used to approximate the solution y r−1 (t) of the given system of integrodifferential equations. This approximate solution will involve Legnedre polynomials of degree r − 1 defined on q subintervals of [0, 1].
This paper is organized as follows. In Section 2, we introduce hybrid functions and its properties.
In Section 3, we describe the method for approximating solutions of the Fredholm and Volterra integro-differential Systems (3) and (4). An upper bound of the error is given in Section 4, and finally numerical results are reported in Section 5.

PRELIMINARIES
In this section, we define the Legendre polynomials p m (t), as well as block-pulse and hybrid functions. We also recall functions' approximation in the Hilbert space The Legendre polynomials p m (t) are polynomials of degree m defined on the interval [−1, 1] by , if m is odd. Equivalently, the Legendre polynomials are given by the recursive formula (see [3,11,17,28]) The block-pulse functions are disjoint and have the property of orthogonality on [0, 1), since for i, j = 1, 2, . . . , q, we have: where ., . is the scalar product given by f, g = 1 0 f (t)g(t)dt, for any functions f, g ∈ L 2 [0, 1).
Definition 2.2. [3,11,17,28] Let r be an arbitrary positive integer. The set of hybrid functions {b km (t)}, k = 1, 2, . . . , q, m = 0, 1, . . . , r − 1, where k is the order for block-pulse functions, m is the order for Legendre polynomials and t is the normalized time, is defined on the interval [0, 1) as Since b km (t) is the combination of Legendre polynomials and block-pulse functions which are both complete and orthogonal, then the set of hybrid functions is a complete orthogonal system in L 2 [0, 1).
We are now able to define the vector function B(t) of hybrid functions on [0, 1) by . . , q, and V T denotes the transpose of a vector V .
Function approximation [3,11,17,28]: Every function f (t) ∈ L 2 [0, 1) can be approximated as Thus, where F is the rq × 1 column vector having f km as entries. In a similar way, any function g(t, s) ∈ can be approximated as where G = (g ij ) is an rq × rq matrix given by Operational matrix of integration [3,11,17,28]: The integration of the vector function B(t) may be approximated by t 0 B(t )dt P B(t), where P is an rq × rq matrix known as the operational matrix of integration and given by where H and E are r × r matrices defined by The integration of two hybrid functions [3,11,17,28]: The integration of the cross product of two hybrid function vectors is given by where D is the r × r matrix given by The matrix ‹ C associated to a vector C [3,15,17,28]: For any rq × 1 vector C, we define the ‹ C is called the coefficient matrix. In [15], Hsiao computed the matrix ‹ C for r = 2 and q = 8, while the authors in [3] considered the case of r = 4 and q = 3.

The vector "
S associated to a matrix S: For any rq × rq matrix S, we define the 1 × rq row vector " S such that B T (t)SB(t) = " SB(t). For instance, let S be a 12 × 12 matrix with coefficients s 11 , s 12 , . . . , s (12) (11) , s (12) (12) . After developing and comparing the two sides of the equation B T (t)SB(t) = " SB(t), we deduce that the 1 × 12 row vector " S is given by:

MAIN RESULTS
In this section, we approximate solutions y(t) of Systems (3) and (4). For this, we need the approximation of y (n) (t).
Lemma 3.1. Let y(t) be a function and consider its approximation If Y (n) denotes the approximation of y (n) (t), then for any n ≥ 1, we have: Approximating y(t), y (t) and y 0 (t), we get Thus, Y T = (Y (1) ) T P + Y T 0 and so Y = P T Y (1) + Y 0 , giving that Y (1) = J(Y − Y 0 ) and the result is true for n = 1.
By induction, assume that the result is true for n and prove it for n + 1. We have which is the desired result.
We are now ready to approximate solutions of Systems (3) and (4). 3.1. Approximated Solution of the Fredholm Integro-Differential System (3). Using the approximations (5) and (6) of functions of one and two variables, System (3) can be approximated as Using Lemma 3.1, the last equation becomes This is a nonlinear system of rq equations in rq variables which can be solved by any iterative method.

Approximated Solution of the Volterra Integro-Differential System (4).
Using the approximations (5) and (6), System (4) can be approximated as: Consider the matrix S := G Y (m) Y (n) P . We obtain Hence, Finally, using Lemma 3.1 for Y (m) and Y (n) , we get a nonlinear system which can be solved by any iterative method.
We recall here that the L 2 norm of a function y : [0, 1] −→ R is given by Theorem 4.1. If y µ (t) = B T (t)Y is the best approximation of the solution y(t) obtained using Legendre polynomials then for some constant M > 0.
Proof. Let X µ be the space of all polynomials of degree less than or equal to µ. Since y µ is the best approximation to y, y − y µ 2 ≤ y − g 2 , for any arbitrary polynomial g in X µ . Therefore by using (9), we get The result is obtained by taking square-root on both sides. 8

NUMERICAL EXAMPLES
In this section, we apply the methods described in Section 3 to some numerical examples to solve Systems (3) and (4). The matrix approximations F of the function f (t) = e t+1 and Y 0 of y(0) are respectively given by From Equation (7), we deduce Using the approximation y 1 (t) = Y T B(t) = B T (t)Y , we get y 1 (t) = 4e − 10 + (18 − 6e)t. In Figure 1, we compare this approximate solution y 1 (t) with the exact solution e t . The absolute errors at various values of t are shown in Table 1. This is degree 1 approximation, therefore µ = 1. Further, M = max [0,1] y (t) = e. The error estimate by using (11) is M/2 4 0.169893. It can be checked from Table 1 that our computed values are less than this error-bound.
Case 2: Now, we consider r = 3 and q = 4. A 12 × 12 matrix K is given by: Other approximations in this case are as below: The approximate solution of (12) is given by  Figure 2 shows the graphs of the approximate solution y 2 (t) and the exact solution e t . The absolute errors at various values of t are given in Table 2. It can be observed that, in this case, the approximate solution is well in agreement with the exact solution. Further, using (11), the error bound is 0.01416. Table 2 shows that our computed maximum value is 0.000145961.  Comparing with the standard form of System (4), we get β = 1, g(t, s) = sin(t − s), m = 0, n = 1, f (t) = 2t 3 + t 2 − 12t + 12 sin(t), l = 0 and a 0 = 0. We take r = 3 and q = 4. Following the procedure described in Section 2, we get Y 0 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) T , and B(t) is same as given in the previous example. Using Equation (8) The approximate solution y 2 (t) is compared with the exact solution t 2 in Figure 3. Table 3 shows the absolute error in the solution at different values of t. Using (11), the error-bound for this example is approximately 0.0104167 and our value from Table  3 is 0.0000271471.

CONCLUSION
In this work, we have discussed an efficient method to solve a class of Fredholm and Volterra integro-differential equations. Our method is based on a new approximation for the derivatives of the equations' solutions using hybrid and block-pulse functions. The absolute errors reported in tables show that the approximate solution is in a good agreement with the exact solution. It is verified in each example that the practical error in our method is less than the theoretical error-bound. In fact, this method is highly efficient, very easy and a powerful mathematical tool for finding the numerical solution of some class of Fredholm and Volterra integro-differential equations. The approximate solutions are found by using the computer code written in Matlab. The method is computationally attractive, and applications are demonstrated through illustrative examples. For future research works, we can use this method to solve various kinds of problems such as higher dimensional problems, stochastic integro-differential equations and partial integro-differential equations of fractional order with additional work.