# Daubechies Wavelet Scaling Function Approach to Solve Volterra’s Population Model

Authors: Amjad Alipanah; Korosh Arzideh; Medina Firouzi; A. Kasnazani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this paper, we focus on a collocation approach based on Daubechies wavelet scaling functions for approximating the solution of Volterra’s model of population growth of a species with a closed system. We present that the integral and derivative terms, which appear in Volterra’s model of the population, will be computed exactly in dyadic points. Utilizing this collocation technique, Volterra’s population model reduces into a system of nonlinear algebraic equations. In addition, an error bound for our method will be explored. The numerical results demonstrate the applicability and accuracy of our method.


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## Body

## 1. Introduction

Daubechies wavelet scaling functions have become a popular method for applications in signal analysis and approximation[1–6]. The Daubechies wavelet and their scaling functions have some features such as orthogonality, vanishing moment, and compact support, and applying them in numerical analysis makes sense. The main issue with Daubechies scaling functions is the lack of closed form, and their value is not known except in dyadic points. Despite this issue, because of their refinement relation, we are able to compute their derivation and integral in dyadic intervals accurately [7]. Adding this feature to their previously mentioned remarkable properties makes the result obtained with them reliable and more accurate than other methods.Volterra’s population model for population growth of a species within a closed system is given [8, 9].(1)dpdt˜=ap−bp2−cp∫0t˜pxdx,p0=p0,where positive constants a,b and c are the birthday rate, the crowding coefficient, and the toxicity coefficient, respectively. p0 is the initial population and pt˜ denotes the population time t˜. The additional integral term indicates the accumulated toxicity of the species. Dimensionless variables are introduced as follows by applying the scale and time population:(2)t=t˜cb,u=pba,to obtain the nondimensional problem,(3)κdudt=u−u2−u∫0tuxdx,u0=u0,where ut is the scaled population of identical individuals at the time t. The trivial solution for equation (3) is ut=0, and the nontrial analytical solution given in [10] shows that ut>0 for all t if u0>0.Several numerical and analytical approximations for solving equation (3) have been reported such as the successive approximation method [8], singular perturbation method [9], Euler and Runge Kutta methods [10]. In [11], a domian decomposition method and the sinc-Galerkin method were compared, and in [12], the series solution method, decomposition method, and Pade approximation were compared. The spectral methods were also applied to solve equation (3) in [13‐15]. In [16], the rational Chebyshev functions and the Hermite function collocation approach are used to solve equation (3). Equation (3) in [16]is converted to a non-linear ODE. In most of these methods (see [11, 17]), for computing the integral in (3), quadrature approximation methods are used, but in our method, we compute it exactly.This paper is organized as follows: in Section2, multiresolution analysis and scaling functions of the Daubechies wavelet are introduced. In Section 3, computations of the integral and the derivation of the scaling functions of the Daubechies wavelet are illustrated. In Section 4, the application of these scaling functions to construct a numerical method for solving (2) is extended, and error analysis is investigated. Finally, in Section 5, the numerical results and the conclusion are presented.


## 2. Multiresolution Analysis and Daubechies Wavelets

Multiresolution analysis is applied to represent a given functionf into different scale components [4, 5]. In order to do this, the basis function ϕ that can be scaled to give multiple resolutions of the original function is introduced. In order to achieve representation of functions in the Hilbert space L2ℝ at different levels of resolution, a nested sequence of closed subspaces Vj was sought as shown in the following equation:(4)Vj⊂Vj+1,j∈ℤ,with the following properties [2, 5]:(1).
∪j∈ℤVj¯=L2ℝ(2).
∩j∈ℤVj=0(3).
fx∈Vj⇔f2x∈Vj+1,j∈ℤ(4).
fx∈V0⇔fx−k∈V0,k∈ℤ(5).
There existsϕ∈V0 in which the set ϕx−k:k∈ℤ is an orthonormal basis of V0Then, these properties ϕj,kx=2j/2ϕ2jx−k form an orthonormal basis for the space Vj. For each j, since Vj is a proper subspace of Vj+1, there is some space in Vj+1 called Wj, which when combined with Vj gives us Vj+1. This space Wj is called the wavelet subspace and is orthogonal complementary to Vj in Vj+1. In other words,(5)Vj∩Wj=0,j∈ℤ,Vj⊥Wj,j∈ℤ,and so(6)Vj+1=Vj⊕Wj,where ⊕ represents a direct sum. It follows, then, we have(7)L2ℝ=⊕j∈ℤWj.Now, let us introduce a wavelet functionψx that ψx−k forms an orthonormal basis of the subspace W0, i.e., W0=spanψx−k:k∈ℤ. Then, we obtain(8)ψj,kx=2j/2ψ2jx−k,j,k∈ℤ,which is an orthonormal basis for Wj. Furthermore, since W0 and V0 are contained in the space V1, we can express the scaling function ϕx and the wavelet function ψx in terms of the scaling function at the next higher scale as shown in the following equations:(9)ϕx=∑k∈ℤhkϕ2x−k,(10)ψx=∑k∈ℤgkϕ2x−k.Equation (9) is referred to as the dilation equation, and the parameters hk are known as filter coefficients. Equation (10) is referred to as the wavelet equation, and the parameters gk are known as wavelet parameters.


### 2.1. Daubechies Scaling Coefficients

In order to determine scaling coefficients, Daubechies imposed some useful conditions as follows:(1)
A scaling function must be able to exactly represent polynomials of order smaller thanp.(2)
∫02p−1ϕxdx=1.(3)
Orthogonality of a scaling function to its integer translates and orthogonality of a scaling function to the Daubechies wavelet are shown in the following equations:(11)∫02p−1ϕx−kϕx−ldx=δk,ll,k∈ℤ,∫02p−1ϕxψx−ldx=0,l∈ℤ.These conditions conclude the following results(12)∑k=0D−1−1kkmhk=0,m=0,1,…,p−1,(13)∑k=02p−1hk=2,(14)∑k=0D−1hkh2m+k=2δ0,m,k=1−p,…,p−1,(15)hk=0,∀k∉0,1,…,2p−1.Imposing equations (12-15) to filter coefficients enables us to compute hkk=02p−1, so dilation equation (9) for the Daubechies scaling function will be as follows:(16)ϕx=∑k=02p−1hkϕ2x−k.In [18], the exact values of hk are evaluated. In [19], it is proved that for a fixed p, there exists only one linear independent scaling function ϕ which satisfies the dilation equation with these filter coefficients. The support of ϕ lies in 0,2p−1, and if we denote D=2p, then the parameter D is called the wavelet genus or Daubechies’ number [1, 6]. The relation between hk and gk is shown in the following equation [5]:(17)gk=−1khD−k−1.In Table1, filter coefficients of the Daubechies scaling function with p vanishing moments have been shown.Table 1
Filter coefficients of the Daubechies scaling function.
khkp=200.683012701892219311.183012701892219220.31698729810778073−0.1830127018922192p=300.470467207784163611.141116915831443620.65036500052623233−0.19093441556832744−0.120832208310396350.0498174997368837p=400.325803428051298211.010945715091828620.89220013826090153−0.03957502623564474−0.264507167369039850.043616300474177260.04650360107098187−0.0149869893303614p=500.226418982583558410.853943542705028321.024326944259196730.19576696134780874−0.34265671538293535−0.045601131883546960.10970265864213397−0.00882680010835838−0.017791870101954290.0047174279390679p=600.157742432002746610.699503814077423321.062263759880189030.44583132293117024−0.31998659894099835−0.183518064106593860.137888092978530470.03892320970789708−0.044663748305460190.0007832511506546100.006756062361590711−0.0015235338263795p=700.110099430745616010.560791283625488221.031148491636141530.66437248221107354−0.20351382246263065−0.316835011280617960.100846465009383970.11400344515973518−0.05378245258968529−0.0234399415642046100.0177497923793598110.000607514995402212−0.0025479047181871130.0005002268531225


### 2.2. Computing Daubechies Scaling Functions and Their Derivative at Integers and Dyadic Points

Generally, scaling functions of the Daubechies wavelet do not have a closed form. Instead, they have to be generated recursively from dilation equation (10). The scaling function ϕ has support for the interval 0,D−1. Putting x=0,…,D−1 in (10) yields a homogeneous linear system of equations. For the Daubechies scaling functions, we have ϕ0=0 and ϕD−1=0, and so, we will discard ϕD−1 in our computations, but for reasons which are explained below, we keep ϕ0. In a matrix form, it becomes(18)h000…00h2h1h0…00h4h3h2…00………………000…hD−3hD−4000…hD−1hD−2ϕ0ϕ1ϕ2…ϕD−3ϕD−2=ϕ0ϕ1ϕ2…ϕD−3ϕD−2,Mϕ=ϕ.Thus, the vector of integer values of the scaling functionϕ is the eigenvector of M corresponding to the eigenvalue 1. As in all eigenvalue problems, the solution to the following system:(19)M−Iϕ=0,for obtaining ϕ, the solution is not unique. In order to obtain a unique solution, the normalizing condition is as follows:(20)∑k=0D−1ϕk=1,which will be imposed on (19) [4]. So, by considering ϕ0=ϕD−1=0, from (20),we have(21)∑k=1D−2ϕk=1.Therefore, from (20) and (21), we obtain the following equation:(22)Mnewϕnew=e1,where Mnew is(23)111…11h2h1−1h0…00⋮⋮⋮⋱⋮⋮000…hD−3−1hD−4000…hD−1hD−2−1,ϕnew=ϕ0ϕ1…ϕD−3ϕD−2,e1=10…00.The values of the scaling function at the integers are given by the solution to equation (22). Now, we can use (16) to obtain the scaling function at all midpoints between integers in the interval, and the value of the Daubechies scaling function in all dyadic points will be determined. The complete explanation of this procedure is named the cascade algorithm and is introduced in [6, 17].We applied the same procedure as the cascade algorithm for evaluating the scaling function’s derivatives at dyadic points. Differentiating equation (16) m times yields(24)ϕmx=2m∑k=0N−1hkϕm2x−k,and by substituting all integer values of the interval 0,D−1, equation (24) gives a linear homogeneous system. The coefficient matrix of this homogeneous system is singular. Thus, a normalization condition is required, in order to determine a unique solution. The following important additional property of the Daubechies scaling function ϕ can be used [20]:(25)∑kkmϕx−k=xm+∑k=1m−1km!m−k!k!xm−k∫−∞∞ϕzzkdz,where m is a positive integer number. By differentiating m times, the previous equation yields(26)∑kkmϕmx−k=m!.Using this normalizing equation (26) and equation (24) for x=1,2,…,D−1 yields an inhomogeneous system, which can be solved, and derivatives can be evaluated at the integer values of x. Similar to the cascade algorithm, using the dilation equation once again, the values of ϕmx at x=i2−n for i=1,3,5,…,D−12n−1 can be determined.


## 2.1. Daubechies Scaling Coefficients

In order to determine scaling coefficients, Daubechies imposed some useful conditions as follows:(1)
A scaling function must be able to exactly represent polynomials of order smaller thanp.(2)
∫02p−1ϕxdx=1.(3)
Orthogonality of a scaling function to its integer translates and orthogonality of a scaling function to the Daubechies wavelet are shown in the following equations:(11)∫02p−1ϕx−kϕx−ldx=δk,ll,k∈ℤ,∫02p−1ϕxψx−ldx=0,l∈ℤ.These conditions conclude the following results(12)∑k=0D−1−1kkmhk=0,m=0,1,…,p−1,(13)∑k=02p−1hk=2,(14)∑k=0D−1hkh2m+k=2δ0,m,k=1−p,…,p−1,(15)hk=0,∀k∉0,1,…,2p−1.Imposing equations (12-15) to filter coefficients enables us to compute hkk=02p−1, so dilation equation (9) for the Daubechies scaling function will be as follows:(16)ϕx=∑k=02p−1hkϕ2x−k.In [18], the exact values of hk are evaluated. In [19], it is proved that for a fixed p, there exists only one linear independent scaling function ϕ which satisfies the dilation equation with these filter coefficients. The support of ϕ lies in 0,2p−1, and if we denote D=2p, then the parameter D is called the wavelet genus or Daubechies’ number [1, 6]. The relation between hk and gk is shown in the following equation [5]:(17)gk=−1khD−k−1.In Table1, filter coefficients of the Daubechies scaling function with p vanishing moments have been shown.Table 1
Filter coefficients of the Daubechies scaling function.
khkp=200.683012701892219311.183012701892219220.31698729810778073−0.1830127018922192p=300.470467207784163611.141116915831443620.65036500052623233−0.19093441556832744−0.120832208310396350.0498174997368837p=400.325803428051298211.010945715091828620.89220013826090153−0.03957502623564474−0.264507167369039850.043616300474177260.04650360107098187−0.0149869893303614p=500.226418982583558410.853943542705028321.024326944259196730.19576696134780874−0.34265671538293535−0.045601131883546960.10970265864213397−0.00882680010835838−0.017791870101954290.0047174279390679p=600.157742432002746610.699503814077423321.062263759880189030.44583132293117024−0.31998659894099835−0.183518064106593860.137888092978530470.03892320970789708−0.044663748305460190.0007832511506546100.006756062361590711−0.0015235338263795p=700.110099430745616010.560791283625488221.031148491636141530.66437248221107354−0.20351382246263065−0.316835011280617960.100846465009383970.11400344515973518−0.05378245258968529−0.0234399415642046100.0177497923793598110.000607514995402212−0.0025479047181871130.0005002268531225


## 2.2. Computing Daubechies Scaling Functions and Their Derivative at Integers and Dyadic Points

Generally, scaling functions of the Daubechies wavelet do not have a closed form. Instead, they have to be generated recursively from dilation equation (10). The scaling function ϕ has support for the interval 0,D−1. Putting x=0,…,D−1 in (10) yields a homogeneous linear system of equations. For the Daubechies scaling functions, we have ϕ0=0 and ϕD−1=0, and so, we will discard ϕD−1 in our computations, but for reasons which are explained below, we keep ϕ0. In a matrix form, it becomes(18)h000…00h2h1h0…00h4h3h2…00………………000…hD−3hD−4000…hD−1hD−2ϕ0ϕ1ϕ2…ϕD−3ϕD−2=ϕ0ϕ1ϕ2…ϕD−3ϕD−2,Mϕ=ϕ.Thus, the vector of integer values of the scaling functionϕ is the eigenvector of M corresponding to the eigenvalue 1. As in all eigenvalue problems, the solution to the following system:(19)M−Iϕ=0,for obtaining ϕ, the solution is not unique. In order to obtain a unique solution, the normalizing condition is as follows:(20)∑k=0D−1ϕk=1,which will be imposed on (19) [4]. So, by considering ϕ0=ϕD−1=0, from (20),we have(21)∑k=1D−2ϕk=1.Therefore, from (20) and (21), we obtain the following equation:(22)Mnewϕnew=e1,where Mnew is(23)111…11h2h1−1h0…00⋮⋮⋮⋱⋮⋮000…hD−3−1hD−4000…hD−1hD−2−1,ϕnew=ϕ0ϕ1…ϕD−3ϕD−2,e1=10…00.The values of the scaling function at the integers are given by the solution to equation (22). Now, we can use (16) to obtain the scaling function at all midpoints between integers in the interval, and the value of the Daubechies scaling function in all dyadic points will be determined. The complete explanation of this procedure is named the cascade algorithm and is introduced in [6, 17].We applied the same procedure as the cascade algorithm for evaluating the scaling function’s derivatives at dyadic points. Differentiating equation (16) m times yields(24)ϕmx=2m∑k=0N−1hkϕm2x−k,and by substituting all integer values of the interval 0,D−1, equation (24) gives a linear homogeneous system. The coefficient matrix of this homogeneous system is singular. Thus, a normalization condition is required, in order to determine a unique solution. The following important additional property of the Daubechies scaling function ϕ can be used [20]:(25)∑kkmϕx−k=xm+∑k=1m−1km!m−k!k!xm−k∫−∞∞ϕzzkdz,where m is a positive integer number. By differentiating m times, the previous equation yields(26)∑kkmϕmx−k=m!.Using this normalizing equation (26) and equation (24) for x=1,2,…,D−1 yields an inhomogeneous system, which can be solved, and derivatives can be evaluated at the integer values of x. Similar to the cascade algorithm, using the dilation equation once again, the values of ϕmx at x=i2−n for i=1,3,5,…,D−12n−1 can be determined.


## 3. The Integrals of the Daubechies Scaling Function on Dyadic Intervals

As previously mentioned, the support of the Daubechies scaling function withp vanishing moments is 0,D−1 where D=2p. In this section, first, the integral of the Daubechies scaling function on integer intervals is computed exactly. Then, we use a recursion relation, in order to obtain the integral of the Daubechies scaling function, on dyadic intervals, i.e, on the intervals k2−n,k+12−n, in arbitrary resolution j will be computed [3].


### 3.1. The Integrals of the Daubechies Scaling Function on Integer Intervals

We defineAa,b, which is as follows:(27)Aa,b=∫abϕxdx,and using dilation equation (16) to the scaling function ϕ, then the following result is obtained:(28)A0,i=∫0i∑k=0D−1hkϕ2x−kdx,i>0=12∑k=0D−1hk∫−k2i−kϕtdt=12∑k=0D−1hkA0,2i−k,in other words,(29)A0,i=12∑k=0D−1hkA0,2i−k.From equation (17), it is clear that(30)A0,i=1,i≥D−1,(31)A0,i=0,i≤0.Equations (29)–(31) can be written as a sparse linear system of equations Cx=b. The entries of matrices C, x and b are given as(32)Ci,j=cl2−hi,for$i=j$,−h2i−j,for$i≠j$,bi=cl0,i≤p−1,∑k=02i−p+1hk,i≥p,xi=A0,i,1≤i≤D−2.It is obvious that the matrix of coefficients in these linear systems is sparse and also diagonally dominant, so is nonsingular. For example, the linear system of equationsC when D=4 and D=6 is given as follows:(33)D=4cc2−h1−h0−h32−h2A0,1A0,2=0h0+h1.(34)D=4cccc2−h1−h000−h32−h2−h1−h0−h5−h42−h3−h200−h52−h4A0,1A0,2A0,3A0,4=00h0+h1∑i=03hi.In this case,D=4, the answers are(35)A0,1=0.8496793685588857,A0,2=1.016346035225553.


### 3.2. Integrals of the Daubechies Scaling Function on Dyadic Intervals

In the previous subsection, the values ofAa,b, when a and b are integers, have been computed. Now, by using dilation equation (10), a recursion relation for j∈ℕ and 0≤l≤D−12j−1 will be introduced as(36)Al2−j,l+12−j=∫l2−jl+12−jϕxdx=∫l2−jl+12−j∑k=0D−1hkϕ2x−kdx=∑k=0D−112hk∫l21−j−kl+121−j−kϕtdt=12∑k=0D−1hkAl21−j−k,l+121−j−k.By using this recursion relation, the values ofAa,b, when a and b are dyadic points, are computed. In Table 2, the values of Aa,b of the Daubechies scaling function, when D=4 and a and b are dyadic points, are given.Table 2
The valuesAa,b for D=4 and j=2.
A01/4=9.9095205×10−2A15/4=1.7456594×10−1A29/4=−2.3661145×10−2A1/42/4=1.9107569×10−1A5/46/4=5.3104960×10−2A9/410/4=5.8193442×10−3A2/43/4=2.4938899×10−1A6/47/4=−1.0216272×10−3A10/411/4=1.6326381×10−3A3/41=3.1011948×10−1A7/42=−5.9982607×10−2A11/43=−1.3687209×10−4


## 3.1. The Integrals of the Daubechies Scaling Function on Integer Intervals

We defineAa,b, which is as follows:(27)Aa,b=∫abϕxdx,and using dilation equation (16) to the scaling function ϕ, then the following result is obtained:(28)A0,i=∫0i∑k=0D−1hkϕ2x−kdx,i>0=12∑k=0D−1hk∫−k2i−kϕtdt=12∑k=0D−1hkA0,2i−k,in other words,(29)A0,i=12∑k=0D−1hkA0,2i−k.From equation (17), it is clear that(30)A0,i=1,i≥D−1,(31)A0,i=0,i≤0.Equations (29)–(31) can be written as a sparse linear system of equations Cx=b. The entries of matrices C, x and b are given as(32)Ci,j=cl2−hi,for$i=j$,−h2i−j,for$i≠j$,bi=cl0,i≤p−1,∑k=02i−p+1hk,i≥p,xi=A0,i,1≤i≤D−2.It is obvious that the matrix of coefficients in these linear systems is sparse and also diagonally dominant, so is nonsingular. For example, the linear system of equationsC when D=4 and D=6 is given as follows:(33)D=4cc2−h1−h0−h32−h2A0,1A0,2=0h0+h1.(34)D=4cccc2−h1−h000−h32−h2−h1−h0−h5−h42−h3−h200−h52−h4A0,1A0,2A0,3A0,4=00h0+h1∑i=03hi.In this case,D=4, the answers are(35)A0,1=0.8496793685588857,A0,2=1.016346035225553.


## 3.2. Integrals of the Daubechies Scaling Function on Dyadic Intervals

In the previous subsection, the values ofAa,b, when a and b are integers, have been computed. Now, by using dilation equation (10), a recursion relation for j∈ℕ and 0≤l≤D−12j−1 will be introduced as(36)Al2−j,l+12−j=∫l2−jl+12−jϕxdx=∫l2−jl+12−j∑k=0D−1hkϕ2x−kdx=∑k=0D−112hk∫l21−j−kl+121−j−kϕtdt=12∑k=0D−1hkAl21−j−k,l+121−j−k.By using this recursion relation, the values ofAa,b, when a and b are dyadic points, are computed. In Table 2, the values of Aa,b of the Daubechies scaling function, when D=4 and a and b are dyadic points, are given.Table 2
The valuesAa,b for D=4 and j=2.
A01/4=9.9095205×10−2A15/4=1.7456594×10−1A29/4=−2.3661145×10−2A1/42/4=1.9107569×10−1A5/46/4=5.3104960×10−2A9/410/4=5.8193442×10−3A2/43/4=2.4938899×10−1A6/47/4=−1.0216272×10−3A10/411/4=1.6326381×10−3A3/41=3.1011948×10−1A7/42=−5.9982607×10−2A11/43=−1.3687209×10−4


## 4. Solution of Volterra’s Population Model by the Scaling Function of the Daubechies Wavelet

In this section, we propose our method to approximate the solution of equation (3). Let U be the approximate solution of u as(37)ux≅Ujx=∑k=k1k2aj,kϕj,kx.As we know, Volterra’s population model is defined on the interval0,+∞ and decays rapidly [8]. The support of Daubechies scaling functions with N vanishing moment is 0,D−1, where D=2N, so we let k1 and k2 be such that, in which the last point in support of ϕj,k1 is smaller than 0 and the first point in support of ϕj,k2 is bigger than D−1; therefore, we have(38)k1=2−D,k2=2jD−1−1.By integrating equation (37), we can get the following result:(39)∫0tUjxdx=∑k=k1k2aj,k∫0tϕj,kxdx.Therefore, suppose that(40)∫0tUxdx=∑k=k1k2aj,kHj,kt,where(41)Hj,kt=2j/2∫0tϕ2jx−kdx,Hj,kt=2−j/2A−k,2jt−k.In order to approximate u′t in equation (3) by differentiating equation (37), we have(42)Uj′x=∑k=k1k2aj,kϕj,k′x.By substituting equations (37)–(42), in equation (3), we construct the residual as(43)Rt=κdUjtdt−Ujt+Uj2t+Ujt∫0tUjxdx,Uj0=u0,and by substituting equation (40) in equation (43), we have(44)Rt=∑k=k1k2aj,kκϕj,k′t−ϕj,kt+ϕj,kt∑k=k1k2aj,kϕj,kt+ϕj,kt∑k=k1k2aj,kHj,kt.In the above equation for obtaining the coefficientsaj,k, let(45)Rtj=0,wherej=1,2…,k2−k1,and also, the initial condition is(46)∑k=k1k2aj,kϕj,k0=u0,in which the collocation points tj are dyadic points on 0,D−1. The equations. (45) and (46) are a system of nonlinear k2−k1+1 equation, which can be solved by Newton’s iteration method.


## 5. Error Analysis

LetN be the nonlinear operator, then we have(47)Nu=κu′t−ut+u2t+ut∫0tuxdx,and it is obvious that Nu=0 when ut is the exact solution of (47). Let Ujt=∑k=k1k2aj,kϕj,k, where j is the resolution of the Daubechies wavelet, then the error of our method is(48)Ej=NUj.LetUjt be the approximate solution of (3) given by (37), and the error of our method is given in (48). Then, the error is(49)Ej=O2−j.Proof.
From equation (47) and Nu=0, we have(50)Ej=NUj=NUj−Nu=κu′−Uj′+u−Uj+u2−Uj2+ut∫0tuxdx−Ujt∫0tUjxdx.Therefore, we have(51)Ej≤κu′−Uj′+u−Uj+u−Uju+Uj+ut∫0tuxdx−Ujt∫0tUjxdx,
and by using the bound of errors asut−Ujt≤C12−j and u′t−Uj′t≤C22−j, which are given in [21] and also C1 and C2 are constants, we have(52)Ej≤κC22−j+C12−j+2C3C12−j+ut∫0tuxdx−UN,jt∫0tUjxdx.
Now, letUjt=ut+εt, then the above inequality can be written as(53)Ej≤C1+κC2+2C3C12−j+εt∫0tεxdx+ut∫0tεxdx+εt∫0tuxdx,and as we know εt≤C12−j, t≤L−1 and if we let C=C1+κC2+2C3C1, then, we have(54)Ej≤C2−j+L−1C122−2j+2C3C1L−12−j,hence(55)Ej=O2−j.
This theorem indicates that the error of our method is bounded, and whenj increases, then the error of our method tends to zero, in the other words, Ej=O2−j, and the numerical results in Table 3 are consistent with this upper bound.Table 3
A comparison between our method at the dyadic point with the exact value.
κExactumaxDyadicumax in our methodj=6j=7j=80.020.923427170.954275220.922526980.923324780.10.769741490.769213120.769745630.769741450.20.659050380.659076930.659051640.659050370.50.485190300.485191310.485190300.48519030


## 6. Numerical Results

In this section, we use the Daubechies scaling function with 4 vanishing moments, and the computational results are summarized in Table3. In Figure 1, the results showed for κ=0.02,0.1,0.2 and 0.5. This figure shows the rapid rise along the logistic curve followed by the slow exponential decay after reaching the maximum point, and when t increases, the amplitude of ut decreases, whereas the exponential decay increases. All computational efforts in this work have been carried out using Maple software in 30 decimal digits.Figure 1
The result of Daubechies wavelet scaling functions forκ=0.02,0.1,0.2,0.5.


## 7. Conclusion

In this paper, we have constructed an efficient and accurate numerical method based on Daubechies wavelet scaling functions for the solution of Volterra’s population model. Our method was used in a direct way without using linearization, perturbation, or restrictive assumption. The refinement equation in the Daubechies wavelet scaling function has enabled us exactly to compute the integral and the derivative of unknown functions in 2. Essential data have been saved, and this fact strongly has reduced time-consuming computations and gives us the advantage of requiring a shorter time to run the procedure in more accurate resolutions. We have shown that our method is simple. The only error that affected the results was computations in the Newton method for solving a system of nonlinear equations. In the error analysis, we showed that when finer resolutions applied for bigj, the error decreases, and numerical results satisfy this result. The stability and convergence of Daubechies wavelet scaling functions make this approach very attractive for computation at the dyadic points. Considering this exact computation and helpful characteristics of Daubechies scaling functions, promising results were obtained.



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