NUMERICAL SOLUTION OF THE VOLTERRA-FREDHOLM INTEGRAL EQUATION BY THE FRAMELET METHOD

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, a numerical method is proposed to solve the Volterra-Fredholm Integral Equation. The method is based on the use of framelets to reduce the integral equation to a system of algebraic equations, a numerical solution is given by using collocation method. Finally, numerical examples are presented to test the efficiency of the proposed method. Comparative results show that our method is more accurate than existing ones.


INTRODUCTION
Framelets have been widely researched in literature and have been successfully extended to many applications, for example, the redundant representation offered by the tight framelet, made it a beneficial tool in signal processing [1] and image compression and restoration [2].
Several articles have shown that constructing and building tight framelets, is much simpler and more versatile than orthonormal wavelet bases [3], [4], [5].
In this paper, tight framelets extracted from refinable functions via a Multiresolution Analysis (MRA) are of special interest to us. The concept of Hilbert space frame was first introduced in 1952 by Duffin and Schaffer [6], where nonharmonic Fourier series has been discussed. The topic has been revived in 1986 by Daubechies, Grossmann,and Meyer in [7] where they gave a solid mathematical basis to the discrete wavelets. Ron and Shen in [8] offered a general characterization of all framelets, concentrating on tight framelets. The authors in [9] discussed frames from a numerical analysis point of view and concluded that truncated frames often result illconditioned linear system. Several articles emphasis on framelets built through MRA and extension principles, as this ensures the presence of fast frames algorithms for application, refer to [10] [11]. MRA symmetric tight framelets resulting from symmetric refinable functions are of interest in both theory and applications.
Many problems that appear to handled with ordinary and partial differential equations can be recast as integral equations, as well as, being useful tool to state many problems in mathematical physics. One of these is Volterra-Fredholm Integral Equation (VFIE) which usually occur from the mathematical modelling of the spatiotemporal growing of some epidemics and from the theory of nonlinear boundary value problems.
The method of separation of variables has been used in [12] to convert (VFIE) of the second kind to the Volterra Integral Equation of the second kind. The author in [13] used the framelets to give a numerical solution of the Volterra Integral Equation. Amin et al. [14] developed a numerical technique to solve the delay (VFIE) by using Haar wavelet. Method based on the two dimensional Legendre wavelets has been presented in [15] to approximate the solution of the Mixed Voltera-Fredholm Integral Equation (MVFIE) . Wazwaz [16] presented a method for solving (MVFIE) using Adomian decomposition series. In [17], biorthogonal tight framelets are used to solve weakly singular (MVFIE).
The standard form of the Volterra-Fredholm integral is given by where u is an unknown function and K 1 (x,t) and K 2 (x,t) are the kernels of the equation.
The paper is organized as follows. In Section 2 we present some concepts and properties of framelets. Then we turn to MRA and the related extension principles, as well as, tight framelets based on B-spline in Section 3. The proposed method is introduced in Section 4. We conclude this paper (Section 5) with some numerical examples to provide the efficiency of the method.

DEFINITIONS AND BASICS
The main purpose of this section is to introduce some basic notations and concepts.
The support of the function f is defined by The definition of frame is what follows: Note that one can extend the functions in a frame for L 2 (a, b) to functions in L 2 (R), by defining them to be zero on R\(a, b) gives a frame for L 2 (R). On the other hand, restricting the functions in a frame for L 2 (R) to the interval (a, b) gives a frame for L 2 (a, b).
For f ∈ L 1 (R), the Fourier transform of f is defined by Given two functions f , g ∈ L 1 (R), the convolution f * g is defined by with the property that f * g(γ) =f (γ)ĝ(γ) ∀γ ∈ R.
Let l 2 (Z) be the set of all sequences h = {h k } defined on Z such that The Fourier series for h ∈ l 2 (Z) is given bŷ The bracket product of two functions f , g ∈ L 2 (R), denoted by [ f , g], and defined by For any closed subset S in L 2 (R) and any function f ∈ L 2 (R), the approximation error is A function φ is said to be refinable if it satisfies the refinement equation for some h 0 ∈ l 2 (Z), called the refinement mask of φ . In Frequency (Fourier) domain, the definition of refinability of φ can be written as Throughout this paper, we assume thatĥ 0 (0) = 1 and all sequences on Z are assumed to be real-valued. Consequently, it follows from (2) thatφ (γ) = ∏ ∞ k=1 h 0 (2 −k γ). Next, the definition of some important operators in L 2 (R) is given: Given a function f ∈ L 2 (R), we define the following operators: Dilation or scaling by a: , for a ∈ R\{0}; and the dyadic scaling operator is given by The following is important relation between the operators: which implies that for j, k ∈ Z For j, k ∈ Z, define ψ j,k (x) = D j T k ψ = 2 j/2 ψ(2 j x − k), ∀ψ ∈ L 2 (R).
If {ψ j,k } j,k∈Z is an orthonormal basis for L 2 (R), then the function ψ is called a wavelet.
, define the dyadic wavelet system or affine system X(Ψ) as where ψ l, j,k = D j T k ψ l . This system is called a wavelet frame (framelet) of L 2 (R) if there exist constants, A, B > 0 such that In the case where X(Ψ) is a tight framelet with bound one, we have the perfect reconstruction which can be written in particular as The elements of Ψ, i.e., ψ 1 , . . . , ψ r are called the generators for the corresponding framelet.
A significant property of a framelet system is the order of vanishing moments. The framelet system has vanishing moments of order m 0 if, for each ψ l ∈ Ψ,ψ l has a zero of order m 0 at the origin.
Let X(Ψ) be a tight frame system, we define the truncated operator Q n by We say that the tight frame system X(Ψ) provides approximation order m 1 , if, for all f in the Shift-invariant spaces are important in construction of applicable framelets. Notice that the affine framelet system X(Ψ) is not shift-invariant. To associate this system with another shiftinvariant system, Ron and Shen in [8] introduced a notation of quasi-affine system.
where ψ q l, j,k is given by The system X q L (Ψ) is a 2 −L shift-invariant system. The approximation order is the same for the affine system and it's corresponding quasi-affine system.
In our discussion below, we need to define the spectrum of the shift-invariant space V 0 as and it is determined only up to a null set. In fact, if φ is compactly supported, then σ

TIGHT FRAMELETS VIA MRA
In this section, we discuss tight framelets designed via MRA. Principally, we provide general basis and specific algorithms for tight framelets, and we discuss how they can used for building B-spline tight framelets. Several explicit examples are presented.

Multiresolution Analysis. Multiresolution Analysis (MRA) were developed by Mallat
and Meyer in 1986 as a general method to generate wavelet orthonormal bases for L 2 (R) of the form {2 j/2 ψ(2 j x − k)} j,k∈Z . Tight framelets that have a (MRA) setup are preferred in applications because this guarantees the existence of fast decomposition and reconstruction algorithms [11].
In this paper, we adopt a MRA setup as proposed in [10], rather than the original setup.
A wavelet system X(Ψ) is said to be MRA based for L 2 (R) if there exists an MRA sequence of subspaces {V j } j∈Z such that the condition Ψ ⊂ V 1 holds. If, in addition, the system X(Ψ) is a frame, we refer to its elements as framelets.
The generator φ of the MRA is known as a refinable function or a scaling function. It has been demonstrated in literature that the approximation properties of a shift-invariant space are related to the order of the zeros ofφ at 2πk, k ∈ Z. Therefore it is important to examine the behavior In this paper, we assume that φ is a compactly supported refinable function generated by a finitely supported refinement mask and satisfy the following: Based on (2), the refinement maskĥ 0 completely determine φ and therefore the underlying MRA.
which can be written in frequency domain aŝ The periodic measurable functionsĥ 1 , . . . ,ĥ r are called wavelet masks or masks.
The MRA provides approximation order m, if, for every f ∈ W m 2 (R), In the following we present some important extensions of the original MRA. The main purpose of these extensions is to construct tight framelets with prescribed properties.

Extension Principles.
The main extensions to MRA are given in this subsection: The Unitary Extension Principle (UEP) and the Oblque Extension Principle (OEP). The aim is to construct functions ψ 1 , . . . , ψ n belonging to V 1 such that {ψ l; j,k } j,k∈Z,l=1,2,...,r forms a tight frame for L 2 (R).
First, let us define the fundamental function Θ by The definition of Θ ensures The following UEP has been introduced and proved by Daubechies et al. [10].
Theorem 3.1. Let φ be a compactly supported refinable function in L 2 (R) with refinement mask h 0 ∈ l 2 (Z). Suppose that there exist a sequence of measurable and essentially bounded functions Then the resulting wavelet system X(Ψ) is a tight frame in L 2 (R), and the fundamental function The UEP is shown to be a very beneficial method to construct compactly supported spline framelets, but the generators have some constraints, e.g. on the number of vanishing moments.
For more details, refer to [18]. In fact, when the number of vanishing moments increases, the framelet representation of one-dimensional piecewise-smooth functions become sparser.
The primary purpose of the following theorem of Oblique Extension Principle (OEP) is to increase vanishing moments of masks derived from a given refinement mask. This principle was first presented and proved in [10] Theorem 3.2. Let φ be a compactly supported refinable function in L 2 (R) with refinement mask h 0 ∈ l 2 (Z). Assume there exists a 2π-periodic function Θ that satisfies the following: (1) Θ is non-negative, essentially bounded, continuous at the origin, and Θ(0) = 1.
The oblique extension principle can be used to obtain spline tight frame system whose truncated framelet system has high approximation order and whose generators have high order vanishing moments. Proof. refer to [10]. Proof. Sinceψ l (γ) =ĥ l (γ)φ (γ/2) andφ (0) = 1, i.e.,ψ l (0) =ĥ l (0), the vanishing moments of ψ l are determined completely by the order of the zero (at the origin) ofĥ l . Hence, X(Ψ) has vanishing moments of order m 0 is equivalent to Next, we show that the approximation order of the MRA provides an upper bound for the approximation order of it's corresponding framelet system.
Theorem 3.5. Let X(Ψ) be an MRA tight frame system. Assume that the system has vanishing moments of order m 0 , and that the refinable function φ provides approximation order m. Then the approximation order of the tight frame system is min{m, 2m 0 }.
Proof. Let X(Ψ) be a tight framelet. Hence, it is satisfy the OEP conditions, and thus It follows that m in Lemma 3.3 is 2m 0 . Explicit expressions for the piecewise linear B 2 (x) and cubic B 4 (x) are given by It is known [10], that in order to construct tight framelets that provide high approximation order, we should choose the fundamental function Θ in OEP as a suitable approximation, at the origin, to 1/|φ | 2 . Hence, if φ = B m , then we should choose Θ as a 2π-periodic function which approximate γ/2 sin (γ/2) 2m at the origin (see [10] for more results and analysis).

Tight Framelets
We now give two examples of framelets: Linear, respectively, cubic spline framelets constructed via Theorem 3.1. Consequently One can verify that all the conditions in UEP hold and, therefor, the system X({ψ 1 , ψ 2 }) is a tight frame for L 2 (R) with two compactly supported symmetric (anti-symmetric) generators (see Figure 1). Although ψ 2 has two vanishing moment,  . Define the masks as followŝ Consequently and The system X({ψ 1 , ψ 2 ψ 3 , ψ 4 }) (see Figure 2) is a tight frame for L 2 (R) that has vanishing moments of order m 0 = 1 since ψ 1 has one vanishing moment. For the refinable function φ = B 4 we have m = 4. The approximation order of the framelet system is min{4, 2} = 2. , Hence, the symmetric framelets are given by (see Figure 3) where ρ = 21 + τ/8.
The conditions of the OEP are satisfied and so the system X({ψ 1 , ψ 2 , ψ 3 }) is a tight frame for L 2 (R) that has vanishing moments of order m 0 = 4. The approximation order of the framelet system is min{4, 8} = 4.

NUMERICAL PROCEDURE
In this section, we present our method which consists of reducing the Voltera-Fredholm Integral Equation to a set of algebraic equations by expanding the unknown function by B-spline framelets with unknown coefficients. The generated system is ill-conditioned so the Moore-Penrose inverse operator method is used to evaluate the unknown coefficients.
Consider the Voltera-Fredholm Integral Equation where u is an unknown function and f ∈ L 2 [0, 1], are explicitly known. According to the proposed method, the approximate solution is given by Substituting Eq.(9) into Eq.(8) yields Which can be rearranged to By producing the collocation points on the interval [0, 1] and putting (11) in Eq.(10) we get Now, we can write Eq. (12) in the form Rewriting Eq. (12) in matrix form leads to The values of the index k and consequently the number of nodes q depend on the support of the refinable function and the domain of the desired function, i.e. u(x). More specifically, by considering the B 2 framelet system, we have to choose k such that 0 < 2 j x − k < 2 for x ∈ (0, 1), j = −n + 1, −n + 2, . . . , n − 1 and hence suitable values of k are −2 n−2 , −2 n−2 + 1, . . . , 2 n−1 − 1. In other words, the matrix M is given by As a consequence of sparseness of the matrix M and the absence of inverse, we computed the unknown coefficients by

NUMERICAL EXPERIMENTS
In this section, several numerical examples have been given to illustrate the efficiency of the proposed method. All computations and plotting are accomplished using Mathematica [19].
Example 5.1. Consider the following Volterra-Fredholm Integral Equation [20] u The exact solution of this equation is u(x) = x 2 . In Table 1, the exact values of the solution at equidistant nodes are compared with its numerical results obtained by the proposed method using B 2 framelets constructed by UEP for n = 3 and n = 4. It can be seen from the table that as n increases the approximation solution converges toward the exact solution.
x i Exact Value For n = 4, Figure 4 shows a good agreement between the exact and approximate solution .  Example 5.2. Consider the following Volterra-Fredholm Integral Equation [21] u(x) = 2 3 x − 1 3 with the exact solution u(x) = x. For n = 3, Table 2 shows the absolute errors at equidistant nodes for the different framelet systems discussed in this paper. The In Table 3 we have compared the absolute errors at equidistant nodes of the proposed method with those from the hybrid orthonormal Bernstein and block puls function (OBH) [22] for n = 3.
The table shows that the performance of our method is better than that of OBH.
For n = 3, Figure 5 shows the absolute errors of the present method via framelets based on  with the exact solution u(x) = e x . Here, we use the framelet system based on B 4 -spline constructed by OEP to determine the approximate solutions. In Table 4 we have compared the absolute errors of the proposed method with those from the Taylor expansion method [23].
The table shows that the performance of our method is better than that of the Taylor expansion method.
x i Taylor Expansion B 4 , OEP

CONCLUSION
In this paper, the linear Volterra-Fredholm Integral Equation is solved by using B-spline based framelets and collocation method. In the proposed technique, the unknown function u(x) is approximated using framelets and the integral equation is converted to a system of algebraic equations. The efficiency of the presented method has been tested through several numerical examples.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.