In problems defined on a semi-infinite domain, rational Chebyshev or Laguerre functions are the generic choices of basis functions in spectral methods. The rationale is that if the solution is oscillatory near the origin, then large number of basis functions may be required to retrieve the spectral accuracy that is not convenient. In this paper, we propose a novel idea that combines Chebyshev wavelets and orthogonal rational functions for solving problems in semi-infinite domain numerically. The semi-infinite domain [0, ∞) is divided into two subdomains [0, α) and [α, ∞). As the basis functions, Chebyshev wavelets are considered in the subdomain [0, α) and a new class of orthonormal rational functions is derived on the semi-infinite subdomain [α, ∞). By constructing the operational matrices of derivative, integral, and product, and implementing the collocation method, the original problem is transcribed to a system of algebraic equations. A key feature of the method is that for a suitably chosen α, it yields good approximations to the solutions that first oscillate and then decay fast as t → ∞. Application of the proposed method to the Volterra’s model for population growth of a species in a closed system is explained. Numerical results show that the discrete solution exhibits exponential convergence as a function of suitable α and the number of collocation points.

1. Introduction

Problems in unbounded domains arise in many disciplines such as fluid dynamics, aerodynamics, quantum mechanics, electronics, astrophysics, biological mathematics, etc. [1, 2]. The analytical solutions to problems in unbounded domains are not readily attainable and thus the need for finding efficient computational algorithms for obtaining numerical solutions arises. In this respect, a variety of options are available. One of the methods is through the use of orthogonal polynomials over unbounded domains, such as Hermite or Laguerre spectral methods [1, 3, 4]. Another direct approach for solving such problems is based on the orthogonal rational approximations, such as rational Chebyshev, rational Legendre, and sinc and radial basis functions [5–9]. An alternative approach is the use of a suitable mapping to transfer infinite domains to finite domains and then applying the standard spectral methods to the transformed problems in finite domains [10, 11]. The domain truncation method is also a suggested method for problems in unbounded domains [12]. Each of these options has its own drawbacks. For instance, in the domain truncation method, the problem is solved only on a large but finite interval [0, L]. In the mapping method and orthogonal rational approximation, when the solution is oscillatory near the origin, large number of basis functions may be required to retrieve the spectral accuracy that is not convenient. Motivated by these deficiencies, we propose a composite wavelet–rational approach that combines the idea behind the domain decomposition and orthogonal rational approximations. Moreover, our new approach is highly suited for functions that first oscillate near the origin and then tend to zero as t goes to infinity.

The Volterra model for population growth of a species within a closed system is described by an integro-differential equation [13, 14] given as follows:

where a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient, c > 0 is the toxicity coefficient, p ₀ is the initial population, and

denotes the population at time

. Note that, the coefficient c indicates the essential behavior of the population evolution before its level falls to zero in the long term. Moreover, the term

represents the effect of toxin accumulation on the species. By employing the following nondimensional time scale and population scale,

we produce the nondimensional problem:

(1)

where u(t) is the scaled population of identical individuals at time t, and κ = c/(a b) is a prescribed nondimensional parameter. Clearly, the only equilibrium solution of Equation (1) is the trivial solution u(t) = 0. In addition, the analytical solution [14],

shows that u(t) > 0 for all t if u ₀ > 0.

The model problem (Equation 1) is a good test problem for investigating the efficiency and accuracy of numerical methods for solving nonlinear Volterra integro-differential equations. Hence, during the past two decades, several numerical methods have been developed for solving problem (Equation 1) and its fractional order version. For numerical methods for the classic Volterra’s population model, we refer the reader to [15–26] and for the fractional order version refer to [27–34]. In most of the mentioned numerical methods, the solution is obtained in a small interval [0, T] or even the obtained solution in the semi-infinite domain [0, ∞) does not have sufficient accuracy.

To remedy these deficiencies, we propose a composite spectral collocation method for solving (Equation 1) which is a combination of the domain truncation and rational approaches. We first divide the semi-infinite domain [0, ∞) into the subdomains [0, α) and [α, ∞), where α is chosen based on the value of κ. Chebyshev wavelets are utilized in the first subdomain and a new kind of rational orthogonal functions are introduced and implemented as basis functions in the second subinterval. Then, a composite function approximation is derived in the semi-infinite domain [0, ∞) and some error estimates are assessed. Next, operational matrices related to Chebyshev wavelets are reviewed and operational matrices of derivative, integral, and product of the new rational functions are derived for the first time. Finally, the collocation method using the shifted Chebyshev–Gauss–Radau points is implemented to approximate the solutions. In the numerical implementation, we show that by choosing a suitable value for α high accurate numerical results can be obtained over the entire interval [0, ∞).

This paper is organized as follows: In Section 2, some mathematical preliminaries are given. In Section 3, the new composite function approximation is introduced, and in Section 4, some error estimates are given. Section 5 is devoted to the operational matrices related to Chebyshev wavelets and the new set of rational functions. In Section 6, we derive our new composite collocation method. Numerical results are given in Section 7, and in Section 8, we provide some conclusions.

2. Preliminaries

In this section, we provide some properties of the Chebyshev wavelets required for our subsequent development. Then, a new class of rational orthonormal functions is derived.

2.1. Chebyshev Wavelets

Wavelets constitute a family of functions defined by dilation and translation of a single function named the mother wavelet. Let a be a dilation parameter and b be a translation parameter that varies continuously. A family of continuous wavelets is given by the following equation:

(2)

A widely used type of wavelets is the Chebyshev wavelet with four parameters k, n, m, and t, denoted by the following equation:

(3)

where m is the degree of Chebyshev polynomials of the first kind and t denotes the normalized time. They are defined on the interval I = [0, α] by the following equation:

(4)

where

(5)

and T _m(t) is the well-known Chebyshev polynomial of degree m [35]. The set of Chebyshev wavelets is an orthogonal set with respect to the weight function,

(6)

on the subinterval

, n = 1, …, 2^k−1. Further, they form a basis for the Banach space

, where w is the weight function of Chebyshev polynomials.

A function

can be expanded in terms of the Chebyshev wavelet basis as follows:

(7)

The wavelet coefficients in Equation (7) are given by

, where

denotes the inner product with respect to the weight function w _n related to shifted Chebyshev polynomials of the first kind on the subinterval I _n. The following theorem adapted from [36] establishes the convergence of the Chebyshev wavelets series.

Theorem 1. convergence: Any function with bounded second derivative on the interval I can be expanded as an infinite sum of Chebyshev wavelets and the series converges uniformly to the function f.

In Section 4, we will prove some results regarding the approximation error of the Chebyshev wavelet expansion, and in Section 5, we will derive the operational matrices of derivative, integration, and product of Chebyshev wavelets.

2.2. A Novel Orthogonal Rational Basis on the Interval [α, ∞)

Orthogonal rational functions, such as rational Jacobi functions, are very useful in approximating decaying functions defined on a semi-infinite interval and their properties have been extensively studied in the literature (refer to [6–9] for more details).

In this study, we are going to construct a novel set of orthogonal rational functions on the semi-inifinite interval J = [α, ∞), denoted by

, that are based on the sequence of rational functions

. Further, we examine their accuracy in approximating the solution of integral equations. For this purpose, we consider the weight function w(t) = 1 and employ the Gram–Schmidt orthogonalization procedure to compute the functions R _m(t). For instance, we have the following equation:

(8)

Notably, the new set of rational functions {R _m(t)} has the property of orthonormality, that is:

In addition, they can be constructed with low computational complexity and evaluate their derivative and integral is an easy task (see Section 5).

3. Composite Function Approximation

For approximating a function f ∈ L ²(0, ∞), Laguerre polynomials [1] or rational Jacobi functions [7] are commonly used in the literature; nonetheless, they are not very suitable for oscillatory solutions. Here, we propose a composite wavelet–rational approximation that is mainly suited for functions that first oscillate and then tend to zero as t tends to infinity. To this end, we first divide the interval [0, ∞) into two subintervals [0, α) and [α, ∞), where α is chosen based on the structure of the underlying function f. Then, we approximate f on the subinterval [0, α) using Chebyshev wavelets and on the subinterval [α, ∞) using the rational functions R _m(t). More precisely, we define the functions

, m = 0, 1, … as follows:

(9)

Next, the function f is approximated as follows:

(10)

where

and

We recall again that the coefficients c _nm and

are given by the following equation:

and

where 〈⋅,⋅〉_w,I denotes the inner product with respect to the weight function w related to shifted Chebyshev polynomials of the first kind on the interval I. Moreover, based on the definition of the basis {Φ(t)}, it is clear that the set of composite functions {Φ(t)} forms a complete basis of L ²[0, ∞).

4. Error Estimates

In this section, we give some estimates for the Chebyshev wavelet approximation error. Let f ^(l) denotes the weak derivative of the function f of order l. The Sobolev norm of integer order s⩾0 with respect to the weight function w on the interval I is given by the following equation:

(11)

According to Canuto et al. [37] (Chapter 5), we have Lemma 1.

Lemma 1. Let , s⩾0 and P _M f be the truncated Chebyshev series of f and w be the Chebyshev weight. Then:

(12)

Moreover, the truncation error in higher order Sobolev norms is as follows:

(13)

where 1 ⩽ q ⩽ s, and the seminorm of

is as follows:

(14)

Clearly, whenever M⩾s − 1:

(15)

Theorem 2 gives an error estimate for the Chebyshev wavelet approximation given in Equation (10) on the interval I = [0, α].

Theorem 2. Let , s⩾1, and h = 2^1−k α be the length of the subinterval I _n. In addition, suppose that be a piecewise function consisting of w _n on each I _n. Then, whenever M⩾s − 1:

(16)

Also, for 1 ⩽ q ⩽ s, if h ⩽ 1:

(17)

and if h > 1:

(18)

Proof. Using the definition of the Sobolev norm, we have the following equation:

Now, for a given function g, if

then it is easy to show that:

Therefore,

Considering q = 0 and employing Equations (12) and (15), we deduce that:

which proves Equation (16).

Furthermore, for 1 ⩽ q ⩽ s and h ⩽ 1, we have the following equation:

where for the second inequality, we used Equation (13) and the proof of Equation (17) is completed. Finally, for 1 ⩽ q ⩽ s and h > 1, we obtain the following equation:

which proves Equation (18).

Remark 1. In Equation (10), the considered basis {R _m(t)} of rational functions on [α, ∞) is a new defined basis and as far as we know, there is no estimation for its approximation error in the literature. However, our numerical results show that this new basis possesses good accuracy for approximating the problem over semi-infinite interval [α, ∞).

5. Operational Matrices

In this section, we first explain the construction of three operational matrices based on Chebyshev wavelets on the interval [0, α). Then, we derive three operational matrices based on the aforementioned rational functions on the interval [α, ∞).

5.1. Chebyshev Wavelet Operational Matrix of Derivative on [0, α)

Theorem 3. Consider the vector function Ψ(t) of Chebyshev wavelets defined in Equation (10). For the first derivative of Ψ(t), we have the following equation:

(19)

where

(20)

and

(21)

(22)

Proof. It is well-known that the derivative of Chebyshev polynomials can be represented in terms of Chebyshev polynomials of lower orders, that is:

(23)

By applying Equation (23) to ψ _nm(t) on the interval [0, α), we get the following equation:

Now, let

, where ψ _n(t) = [ψ _n0(t), …, ψ _{n(M − 1)}(t)], n = 1, 2, …, 2^k−1. Since the vector function ψ _n(t) is zero outside the interval I _n, using the above relation and by induction of M, we can write the following equation:

where the matrix E is given in Equations (21) and (22) and the proof is completed.

5.2. Chebyshev Wavelet Operational Matrix of Integration on [0, α)

Theorem 4. Consider the vector function Ψ(t) of Chebyshev wavelets defined in Equation (10). For the integral of Ψ(t), we have the following equation:

(24)

where Q ₁ is a 2^k−1 M × 2^k−1 M matrix given by the following equation:

(25)

in which, C and S are M × M matrices given by the following equation:

(26)

and

(27)

Proof. In order to express the integral of Ψ(t) in terms of Ψ(t) itself, we must first calculate the moments:

To do so, using a simple calculation, we obtain the following equation:

and for m = 2, …, M − 1,

where

. Again, by considering

where ψ _n(t) = [ψ _n0(t), …, ψ _{n(M − 1)}(t)], n = 1, 2, …, 2^k−1 and using the above relation, induction of M and a simple calculation, we obtain the operational matrix Q ₁ as defined in Equations (25)–(27).

5.3. Chebyshev Wavelet Product Operational Matrix on [0, α)

We give an instance for derivation of the Chebyshev wavelet product operational matrix. For other cases, the same line can be followed.

Lemma 2. Consider the set of Chebyshev wavelets for M = 3, k = 3, and α = 2. Then, one has the following equation:

(28)

where

Proof. Using the definition of Chebyshev wavelets, we have the following equation:

and

The above relations are used to obtain the product Ψ(t)Ψ ^T(t). So, for M = 3, k = 3, and α = 2, we obtain the following equation:

As a result, the operational matrix (Equation 28) is obtained.

5.4. Rational-Based Operational Matrix of Integration on [α, ∞)

Consider the vector function R(t) defined in Equation (10). Let:

(29)

where

is the rational base operational matrix of integration. For computing the elements

, we first compute the value as follows:

Then, we have the following equation:

(30)

For instance, for M = 3, we get the following equation:

(31)

5.5. Rational-Based Operational Matrix of Derivative on [α, ∞)

Let:

(32)

where

is the rational-based operational matrix of derivative. Suppose that

, 0 ⩽ m ⩽ M − 1. The elements

are obtained as follows:

(33)

For instance, for M = 3, we get the following equation:

5.6. Rational-Based Product Operational Matrix on [α, ∞)

Clearly, the product R(t)R ^T(t) has the following general form:

(34)

For instance, for M = 3, we have the following equation:

(35)

where

Remark 2. Deriving a general closed form for the operational matrices Q ₂, D ₂, and P ₂ related to the new set of orthogonal rational functions is not easy. However, as our explanations in Sections 5.4, 5.5, and 5.6 show, using a simple code they can be constructed for any given M.

5.7. Operational Matrices of Composite Wavelet–Rational Approximation

Based on the aforementioned operational matrices and the composite function approximation (Equation 10), we introduce the composite operational matrices of derivative, integral, and product as follows:

(36)

(37)

(38)

where

in which, the matrices D ₁, D ₂, Q ₁, Q ₂, P ₁(t), and P ₂(t) have been introduced in Sections 5.1–5.6.

6. The Solution Method

Consider the Volterra’s population growth model given in Equation (1). Let:

(39)

Then, we have the following equation:

(40)

Substituting Equations (39) and (40) into Equation (1), the following equivalent second-order initial value problem is obtained:

(41)

Approximating v ^″(t) using (Equation 10) results the following equation:

(42)

By twice integrating (Equation 42) over the interval (0, t) and utilizing Equation (37), we arrive at:

and

Next, let the constant vectors U ₁ and U ₂ be such that

and

. Therefore,

(43)

(44)

We then substitute the approximation Equations (42)–(44) into Equation (41) and define the residual function, Res _M(t), as follows:

(45)

where P(t) is the composite operational matrix of product given in Equation (38).

The vector C in Equation (45) contains (2^k−1 + 1)M unknowns that must be determined. To this end, in each subinterval

, n = 1, …, 2^k−1 and in the interval J = [α, ∞) we consider M collocation points and then collocate the residual function at these points. The most appropriate choice for collocation points are the shifted Chebyshev–Gauss–Radau quadrature points given by the following equation:

(46)

and

(47)

where

are the standard Chebyshev–Gauss–Radau points. The algebraic equations for obtaining the unknowns of the vector C come from equalizing R e s(t) to zero at the points (Equations 46 and 47). The resulting system of nonlinear algebraic equations can be solved using Newton’s iterative method. It is worth mentioning that introducing the operational matrices and utilizing them in the composite collocation method provides more numerical stability and reduces the computational complexity.

Below is a summary of steps of solutions for problem (Equation 1):

Step 1. For all values of κ, initialize with α = 2, k = 1, and M = 10.
Step 2. Compute ψ _{n
m}(t), 1 ⩽ n ⩽ 2^k−1, 0 ⩽ m ⩽ M − 1 using Equation (4) and R _m(t), 0 ⩽ m ⩽ M − 1 using Equation (8).
Step 3. For the parameters of Step 1, derive the operational matrices of Section 5.
Step 4. Collocate (Equation 45) at the set of collocation points Equations (46) and (47) and solve the resulting system of nonlinear algebraic equations to obtain initial approximate solution using Equation (10).
Step 5. Using the initial approximate solution of Step 4, for each value of κ choose more appropriate value for α and repeat Steps 2–4 for various values of k and M.

7. Numerical Results

In this section, we examine the mathematical structure of the Volterra’s population growth model (Equation 1). The explicit exact solution to this problem is unavailable. Nevertheless, we know that if u ₀ > 0 then u(t) > 0 for t⩾0. Furthermore, according to Wazwaz [15], we seek to study the rapid growth along the logistic curve that will reach a peak, followed by the slow exponential decay where the solution goes to zeros as t → ∞.

We applied the method of Section 6 and solved Equation (41) for u ₀ = 0.1 and various values of κ. In this model, the exact value of u _max is as follows [14]:

For approximating u _max with high accuracy, the choice of α is crucial. Considering this fact, as mentioned before, for all values of κ, we first solved the model for α = 2 and M = 10 to determine the approximate location of t _critical and then we considered an appropriate value of α for each value of κ. In Table 1, the resulting values of u _max for different values of k, M, and α are compared with the exact solutions. In Table 2, we compare our results with the methods proposed in [15, 18, 19, 31]. The abbreviations in this table are as follows:

Table 1. Summary of u _max for various κ using the present method.

κ	α	k = 2, M = 15	k = 3, M = 15	Exact u _max
0.02		0.9234920	0.923427494	0.9234271720702181
0.04		0.8737205	0.87371998315982	0.8737199831539955
0.1		0.7697414906998	0.76974149070048	0.7697414907005955
0.2	1	0.659050381552355	0.659050381552344	0.6590503815523149
0.5	2	0.485190291409431	0.485190291409428	0.4851902914094209

Table 2. Comparison in the errors of approximating u _max.

κ	MADM-P [15]	HFC [18]	DRBF [19]	HWQ [31]	Present method (k = 3, M = 15)
0.02	1.96 × 10⁻²	1.32 × 10⁻⁷	3.0 × 10⁻¹¹	1.1 × 10⁻¹⁰	3.22 × 10⁻⁷
0.04	—	3.15 × 10⁻⁹	4.6 × 10⁻¹¹	3.4 × 10⁻¹²	5.82 × 10⁻¹²
0.1	4.63 × 10⁻³	7.00 × 10⁻¹⁰	9.4 × 10⁻¹²	7.2 × 10⁻¹²	1.15 × 10⁻¹³
0.2	1.14 × 10⁻³	1.60 × 10⁻⁹	5.2 × 10⁻¹¹	5.7 × 10⁻¹³	2.91 × 10⁻¹⁴
0.5	9.21 × 10⁻⁵	8.50 × 10⁻⁹	9.4 × 10⁻¹²	4.9 × 10⁻¹⁴	7.11 × 10⁻¹⁵

•
MADM-P: modified Adomian decomposition method in conjunction with Padé technique [15].
•
HFC: Hermite functions collocation method [18].
•
DRBF: radial basis functions collocation method [19].
•
HWQ: Haar wavelet quasilinearization method [31].

Tables 1 and 2 demonstrate the high accuracy and convergence of the proposed method in the subinterval [0, α). Moreover, the numerical results are in agreement with the theoretical results established in Theorem 1. In order to show the convergence of the method in the subinterval [α, ∞), we report the values of

in Table 3. Moreover, Figure 1 depicts the approximate solutions for κ = 0.02, 0.04, 0.1, 0.2, 0.5 obtained using the present composite wavelet–rational collocation method.

Table 3. Numerical results of

for various values of M.

κ	α	M = 5	M = 10	M = 15
0.02		1.33 × 10⁻¹	5.49 × 10⁻³	6.77 × 10⁻⁵
0.04		3.42 × 10⁻³	7.62 × 10⁻⁶	8.38 × 10⁻⁹
0.1		7.86 × 10⁻³	9.46 × 10⁻⁷	7.73 × 10⁻¹⁰
0.2	1	4.35 × 10⁻⁴	2.96 × 10⁻⁸	9.64 × 10⁻¹⁰
0.5	2	5.36 × 10⁻⁴	3.45 × 10⁻⁸	1.21 × 10⁻¹¹

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

The results of the present method calculation for κ = 0.02, 0.04, 0.1, 0.2, 0.5 in the order of height.

8. Conclusions

A new composite wavelet–rational approach has been proposed for solving integro-differential equations on semi-infinite intervals. This new method was successfully tested on the Volterra model for population growth of a species in a closed system. We divided the semi-infinite domain of the problem into two parts and employed Chebyshev wavelets for the finite subdomain and a class of orthonormal rational functions for the semi-infinite subdomain. It was shown that this domain decomposition can be effectively made based on the structure of the solution. Appropriate sets of collocation points in each subdomain were considered. This approach is easy to be implemented and possesses the spectral accuracy. Numerical results show the excellent agreement between the approximate and exact values for u _max. Moreover, the new method has superiority over some other numerical methods for semi-infinite calculations such as the Padé approximation, radial basis functions, and HFC methods. As suggestions for future works, the proposed method can be extended to fractional order problems. Also, further works are required to obtain estimation for approximation errors of the new basis of rational functions {R _m(t)}.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This study receives no funding.

Open Research

Data Availability Statement

The data supporting this study’s findings are openly available through the text.

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A Composite Wavelet–Rational Approach for Solving the Volterra’s Population Growth Model Over Semi-Infinite Domain

Abstract

1. Introduction

2. Preliminaries

2.1. Chebyshev Wavelets

2.2. A Novel Orthogonal Rational Basis on the Interval [α, ∞)

3. Composite Function Approximation

4. Error Estimates

5. Operational Matrices

5.1. Chebyshev Wavelet Operational Matrix of Derivative on [0, α)

5.2. Chebyshev Wavelet Operational Matrix of Integration on [0, α)

5.3. Chebyshev Wavelet Product Operational Matrix on [0, α)

5.4. Rational-Based Operational Matrix of Integration on [α, ∞)

5.5. Rational-Based Operational Matrix of Derivative on [α, ∞)

5.6. Rational-Based Product Operational Matrix on [α, ∞)

5.7. Operational Matrices of Composite Wavelet–Rational Approximation

6. The Solution Method

7. Numerical Results

8. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

Figures

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A Composite Wavelet–Rational Approach for Solving the Volterra’s Population Growth Model Over Semi-Infinite Domain

Abstract

1. Introduction

2. Preliminaries

2.1. Chebyshev Wavelets

2.2. A Novel Orthogonal Rational Basis on the Interval [α, ∞)

3. Composite Function Approximation

4. Error Estimates

5. Operational Matrices

5.1. Chebyshev Wavelet Operational Matrix of Derivative on [0, α)

5.2. Chebyshev Wavelet Operational Matrix of Integration on [0, α)

5.3. Chebyshev Wavelet Product Operational Matrix on [0, α)

5.4. Rational-Based Operational Matrix of Integration on [α, ∞)

5.5. Rational-Based Operational Matrix of Derivative on [α, ∞)

5.6. Rational-Based Product Operational Matrix on [α, ∞)

5.7. Operational Matrices of Composite Wavelet–Rational Approximation

6. The Solution Method

7. Numerical Results

8. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

Figures

References

Related

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