New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation

Waleed Mohamed Abd-Elhameed; Mohamed Salem Al-Harbi; Ahmed Gamal Atta

doi:10.1515/nleng-2022-0332

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New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation

Waleed Mohamed Abd-Elhameed , Mohamed Salem Al-Harbi and Ahmed Gamal Atta

Published/Copyright: August 2, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This article is dedicated to propose a spectral solution for the non-linear Fitzhugh–Nagumo equation. The proposed solution is expressed as a double sum of basis functions that are chosen to be the convolved Fibonacci polynomials that generalize the well-known Fibonacci polynomials. In order to be able to apply the proposed collocation method, the operational matrices of derivatives of the convolved Fibonacci polynomials are introduced. The convergence and error analysis of the double expansion are carefully investigated in detail. Some new identities and inequalities regarding the convolved Fibonacci polynomials are introduced for such a study. Some numerical results, along with some comparisons, are provided. The presented results show that our proposed algorithm is efficient and accurate.

Keywords: Fitzhugh–Nagumo equation; convolved Fibonacci polynomials; collocation method; operational matrices; convergence analysis

MSC 2010: 65M60; 11B39; 40A05; 34A08

1 Introduction

In various fields of mathematics and physics, special functions play an important role [1–3]. In particular, the investigations of different sequences of polynomials have increased due to their applications in many disciplines such as number theory, statistics, mathematical physics, and numerical analysis [4–6]. We can classify the sequences of polynomials into orthogonal ones and non-orthogonal ones. In the scope of numerical analysis and approximation theory, the use of orthogonal polynomials is an old practice and continues. There are classes of polynomials, such as the classical orthogonal polynomials, that were extensively used in different applications [7–11]. On the other hand, the use of non-orthogonal polynomials has also increased in many applications [12–15].

Fibonacci polynomials and their related numbers are crucial as they appear in various disciplines such as graph theory, coding theory, and numerical analysis [4]. Furthermore, there are many generalizations and modifications of Fibonacci polynomials. There are two classes of polynomials that generalize, respectively, Fibonacci and Lucas polynomials [16]. Haq and Ali [17] used mixed Lucas and Fibonacci polynomials to treat a two-dimensional Sobolev equation. Özkan and Tassstan [18] introduced certain Gauss Fibonacci polynomials. In the study of Tasyurdu [19], bi-periodic generalized Fibonacci polynomials were introduced.

Spectral methods are among the celebrated numerical methods that are used to obtain numerical solutions to different types of ordinary and partial differential equations (PDEs). For some problems that can be handled using the different versions of spectral methods, one can consult previous studies [2,20]. When compared to other numerical approaches for solving PDEs, spectral methods have a number of benefits. One of these advantages is controlling on improving the accuracy when the number of chosen basis functions is increased. Another advantage is that they have the capability to treat different kinds with various initial and boundary conditions. The pivotal step in applying the spectral method is to express the approximate solutions as a sum of a few selected basis functions that may be orthogonal polynomials or non-orthogonal ones [21–26]. How to pick the appropriate spectral technique to use depends on the type of underlying conditions governed by the differential equation. Galerkin method has some restrictions on the two types of basis functions that we select, see for example, refs [27–32], while tau and collocation methods are more flexible than the Galerkin method, see for example, refs [33–40].

The investigations concerned with the initial and boundary value problems are vital due to their emergence in different engineering, scientific, and technological contexts. Many systems that appear in engineering and physical fields can be described by differential equations governed by initial and boundary conditions. Some of the theorems regarding the existence and uniqueness that are related to some types of these equations can be found in the book of Agarwal [41]. In many cases, the solutions to initial value problems (IVPs) and boundary value problems (BVPs) are not found analytically, so the role of numerical analysis in treating such kinds is essential. Some standard techniques, such as Runge–Kutta method, and finite difference methods may be useful in treating some types of IVPs (see, for example, refs [42,43]). Regarding the BVPs, there are many numerical algorithms that were utilized to treat them [44].

In many areas of mathematics, science, and engineering, nonlinear equations play crucial roles as they describe many phenomena. These equations are more difficult to solve than linear equations. Structural analysis, control systems, and electrical circuits are just a few of the many domains in engineering where nonlinear equations arise. Regarding the majority of the non-linear equations, their analytic solutions are not available, and thus these equations require numerical methods to handle them. Among the important non-linear equations is the Fitzhugh–Nagumo (F–N) equation that models the nerve-impulse propagation. This equation was derived by Fitzhugh [45] and Nagumo et al. [46]. Mathematicians and theoretical biologists have taken an interest in this equation for quite some time. In addition, this equation has applications in circuit theory and population genetics [47]. Regarding the F–N equation, many studies were performed to obtain analytic solutions for this equation. For example, Kawahara and Tanaka [48] obtained some exact solutions to F–N equation. Nucci and Clarkson [49] obtained other solutions. From a numerical point of view, in the study of Bhrawy [50], the equation was handled using the Jacobi–Gauss–Lobatto collocation method. A pseudospectral method was applied in the study of Olmos and Shizgal [51]. A spline method was applied in the study of Shekarabi et al. [52]. The finite difference approach was followed in the study of Namjoo and Zibaei [53]. Another finite difference scheme was followed in the study of Gui [54].

In this article, we are interested in introducing a kind of polynomial, namely, convolved Fibonacci polynomials. This type of polynomial generalizes the celebrated type of Fibonacci polynomials. In fact, for a specific choice of the involved parameter, these polynomials are reduced to Fibonacci polynomials. In addition, we will apply the spectral collocation method to numerically solve the F–N equation. We believe that this type of non-orthogonal polynomial is being used for the first time in numerical analysis to solve some specific kinds of non-linear problems.

The main objectives of this article can be listed as follows:

Introducing the sequence of convolved Fibonacci polynomials that generalize the sequence of Fibonacci polynomials.
Introducing some formulas concerned with the convolved Fibonacci polynomials including their operational matrices of derivatives.
Making use of some theoretical results along with the application of the collocation method to obtain numerical solutions of the F–N equation.
Demonstrating the applicability and efficiency of the proposed algorithm.

This article is structured as follows: Section 2 is devoted to presenting some properties of Fibonacci and convolved Fibonacci polynomials. In Section 3, a new expression for the derivatives of convolved Fibonacci polynomials is established; hence, two operational matrices of derivatives are derived. We propose a spectral solution of the F–N equation based on the application of the collocation method in Section 4. The convergence and error analysis are investigated in depth in Section 5. Some test problems are presented in Section 6. Finally, some conclusions are reported in Section 7.

2 An overview on Fibonacci and convolved Fibonacci polynomials

This section focuses on presenting some fundamental characteristics of the Fibonacci polynomials as well as a generalizing class of polynomials, namely, convolved Fibonacci polynomials.

2.1 A brief overview of Fibonacci polynomials

Using the following recursive formula, we can generate Fibonacci polynomials [4]

F k + 2 ( x ) = x F k + 1 ( x ) + F k ( x ) , F 0 ( x ) = 0 , F 1 ( x ) = 1 , k ≥ 2 .

Note that F k + 1 ( x ) is of degree k .

The analytic form of the Fibonacci polynomials is

F n ( x ) = ∑ j = 0 n − 1 2 n − j − 1 j x n − 2 j − 1 ,

where the notation ⌊ y ⌋ represents the floor function.

2.2 An overview on convolved Fibonacci polynomials

For every complex number m , Wang and Wang [55] introduced the convolved ( a , b ) Fibonacci polynomials CV i a , b , m ( x ) , where a ( x ) and b ( x ) are polynomials with real coefficients. The polynomials CV i a , b , m ( x ) (of degree i ) can be constructed using the generating function listed as follows:

( 1 − a ( x ) t − b ( x ) t 2 ) − m = ∑ i = 0 ∞ CV i a , b , m ( x ) t i .

They can be explicitly expressed as

(1) CV i a , b , m ( x ) = ∑ r = 0 i 2 m + r − 1 r m + i − r − 1 i − 2 r a i − 2 r ( x ) b r ( x ) .

Moreover, they satisfy the following recurrence relation:

i CV i a , b , m ( x ) − ( m + i − 1 ) a ( x ) CV i − 1 a , b , m ( x ) − ( 2 m + i − 2 ) b ( x ) CV i − 2 a , b , m ( x ) = 0 , i ≥ 2 ,

with CV 0 a , b , m ( x ) = 1 , CV 1 a , b , m ( x ) = m a ( x ) .

In this article, we are interested in investigating particular polynomials of CV i a , b , m ( x ) that correspond to the following choices:

(2) a ( x ) = x , b ( x ) = 1 .

Remark 1

With the choices in (2), and m = 1, the polynomials defined in (1) reduce to the well-known Fibonacci polynomials, so these polynomials are generalizations of Fibonacci polynomials. We will denote these polynomials by C i m ( x ) .

The recurrence relation satisfied by C i m ( x ) is

(3) i C i m ( x ) − ( m + i − 1 ) x C i − 1 m ( x ) − ( 2 m + i − 2 ) C i − 2 m ( x ) = 0 , i ≥ 2 .

The analytic form of the convolved Fibonacci polynomials C i m ( x ) is

(4) C i m ( x ) = ∑ r = 0 i 2 ( m ) i − r r ! ( i − 2 r ) ! x i − 2 r .

Alternatively, it can be written as follows:

C i m ( x ) = ∑ k = 0 i a i + k ( m ) i + k 2 k ! i − k 2 ! x k ,

where

(5) a r = 1 , if r even , 0 , otherwise .

The following lemma gives the inversion formula of C i m ( x ) .

Lemma 1

Let m be any real non-positive integer. The inversion formula of C i m ( x ) is given by

(6) x i = i ! Γ ( m ) ∑ r = 0 i 2 ( − 1 ) r ( i + m − 2 r ) Γ ( 1 + i + m − r ) r ! C i − 2 r m ( x ) .

Proof

We will prove the formula by induction. The formula is clearly valid for i = 0 . Now, assume that Formula (6) holds, and we are going to show that the following identity holds:

(7) x i + 1 = ( i + 1 ) ! Γ ( m ) ∑ r = 0 i + 1 2 ( − 1 ) r ( i + m − 2 r + 1 ) Γ ( 2 + i + m − r ) r ! C i − 2 r + 1 m ( x ) .

Multiplying both sides of (6) and applying the recurrence relation (3) in the form:

x C i m ( x ) = i + 1 i + m C i + 1 m ( x ) − 2 m + i − 1 i + 1 C i − 1 m ( x ) ,

yield the following formula:

x i + 1 = i ! Γ ( m ) ∑ r = 0 i 2 ( − 1 ) 1 + r ( − 1 − i + 2 r ) Γ ( 1 + i + m − r ) r ! × C i − 2 r + 1 m ( x ) − i + 2 m − 2 r − 1 1 + i − 2 r C i − 2 r − 1 m ( x ) ,

which can be turned into the following one:

x i + 1 = i ! Γ ( m ) ∑ r = 0 i 2 ( − 1 ) 1 + r ( − 1 − i + 2 r ) Γ ( 1 + i + m − r ) r ! C i − 2 r + 1 m ( x ) − ∑ r = 1 i + 1 2 ( − 1 ) 1 + r ( 1 + i + 2 m − 2 r ) ( r − 1 ) ! Γ ( 2 + i + m − r ) C i − 2 r + 1 m ( x ) .

Some simple computations lead to Formula (7).□

3 Derivatives and operational matrices of derivatives of convolved Fibonacci polynomials

This section develops new expressions for the high-order derivatives of the convolved Fibonacci polynomials. In addition, operational matrices of derivatives for these polynomials have been established. These matrices will play a vital role in the derivation of our proposed collocation algorithm.

Theorem 1

For any positive integers q and i with i ≥ q , the qth-derivative of C i m ( x ) can be expressed as

(8) d q C i m ( x ) d x q = ∑ k = 0 i − q A i , k q C k m ( x ) ,

where

A i , k q = ( − 1 ) i − k − q 2 ( k + m ) a i − k − q i − k + q − 2 2 i − k − q 2 × i + k + 2 m − q + 2 2 q − 1 ,

and a k is defined as in (5).

Proof

To prove Formula (8), we prove its alternative formula:

(9) D q C i m ( x ) = ∑ ℓ = 0 i − q 2 ( − 1 ) ℓ ( i − 2 ℓ + m − q ) × ℓ + q − 1 ℓ ( i − ℓ + m + 1 − q ) q − 1 × C i − q − 2 ℓ m ( x ) .

If we differentiate the analytic formula of C i m ( x ) in (4), we obtain

D q C i m ( x ) = 1 Γ ( m ) ∑ j = 0 i − q 2 Γ ( i − j + m ) j ! ( i − 2 j − q ) ! x i − 2 j − q ,

which gives after applying the inversion formula (6)

D q C i m ( x ) = ∑ j = 0 i − q 2 Γ ( i − j + m ) j ! × ∑ r = 0 i − q 2 − j ( − 1 ) r ( i − 2 j + m − q − 2 r ) r ! Γ ( 1 + i − 2 j + m − q − r ) C i − q − 2 ( j + r ) m ( x ) .

The last formula after some algebraic computations turns into

(10) D q C i m ( x ) = ∑ ℓ = 0 i − q 2 ( i − 2 ℓ + m − q ) ∑ t = 0 ℓ × ( − 1 ) ℓ − t Γ ( i + m − t ) ( ℓ − t ) ! t ! Γ ( 1 + i − ℓ + m − q − t ) C i − q − 2 ℓ m ( x ) .

To find a closed formula for the sum that appears in the last formula, we set

R ℓ , i , q = ∑ t = 0 ℓ ( − 1 ) ℓ − t Γ ( i + m − t ) ( ℓ − t ) ! t ! Γ ( 1 + i − ℓ + m − q − t ) .

Zeilberger’s algorithm serves to find the following recursive formula for R ℓ , i , q

( ℓ + 1 ) ( − ℓ + i + m − 1 ) R ℓ + 1 , i , q + ( − ℓ + i + m − q ) ( ℓ + q ) R ℓ , i , q = 0 , R 0 , i , q = Γ ( i + m ) Γ ( i + m − q + 1 ) ,

whose solution is given by

(11) R ℓ , i , q = ( − 1 ) ℓ ( q ) ℓ Γ ( i + m − ℓ ) ℓ ! Γ ( i + m − q − ℓ + 1 ) .

Inserting (11) into Formula (10), Formula (9) can be obtained.□

Corollary 1

The first derivative of C i m ( x ) can be expressed as

d C i m ( x ) d x = ∑ k = 0 i − 1 h i , k C k m ( x ) , i ≥ 1 ,

where

h i , k = ( − 1 ) i − k − 1 2 ( k + m ) a i − k − 1 .

Corollary 2

The second derivative of C i m ( x ) can be expressed as

d 2 C i m ( x ) d x 2 = ∑ k = 0 i − 2 f i , k C k m ( x ) , i ≥ 2 ,

where

f i , k = 1 4 ( − 1 ) i − k 2 ( k − i ) ( k + m ) ( i + k + 2 m ) a i − k − 2 .

Remark 2

The first and second derivatives of C i m ( x ) can be written in the matrix form as

(12) d C m ( x ) d x = ℋ C m ( x ) ,

(13) d 2 C m ( x ) d x 2 = ℱ C m ( x ) ,

where C m ( x ) = [ C 0 m ( x ) , C 1 m ( x ) , … , C N m ( x ) ] T . Also, ℋ = ( h i , k ) and ℱ = ( f i , k ) are operational matrices of derivatives of order ( N + 1 ) × ( N + 1 ) and the entries of these matrices can be written in the form:

h i , k = ( − 1 ) i − k − 1 2 ( k + m ) a i − k − 1 , if i > k , 0 , otherwise , f i , k = 1 4 ( − 1 ) i − k 2 ( k − i ) ( k + m ) ( i + k + 2 m ) a i − k − 2 , if i > k + 1 , 0 , otherwise .

For example, the matrices ℋ and ℱ take the following forms for N = 6

ℋ = 0 0 0 0 0 0 0 m 0 0 0 0 0 0 0 m + 1 0 0 0 0 0 − m 0 m + 2 0 0 0 0 0 − m − 1 0 m + 3 0 0 0 m 0 − m − 2 0 m + 4 0 0 0 m + 1 0 − m − 3 0 m + 5 0 ,

ℱ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 m ( 2 m + 2 ) 0 0 0 0 0 0 0 1 2 ( m + 1 ) ( 2 m + 4 ) 0 0 0 0 0 − m ( 2 m + 4 ) 0 1 2 ( m + 2 ) ( 2 m + 6 ) 0 0 0 0 0 − ( m + 1 ) ( 2 m + 6 ) 0 1 2 ( m + 3 ) ( 2 m + 8 ) 0 0 0 3 2 m ( 2 m + 6 ) 0 − ( m + 2 ) ( 2 m + 8 ) 0 1 2 ( m + 4 ) ( 2 m + 10 ) 0 0 .

It is worth mentioning here that the special structure of matrices ℋ and ℱ makes the evaluation of the derivatives easy.

4 Spectral solution of the F–N equation

This section proposes the numerical scheme designed for solving the F–N equation based on the application of the spectral collocation algorithm.

In this section, we consider the following F–N equation [52]:

(14) χ t = χ x x − χ ( α − χ ) ( 1 − χ ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

χ ( x , 0 ) = u 0 ( x ) , 0 < x ≤ 1 , χ ( 0 , t ) = u 1 ( t ) , χ ( 1 , t ) = u 2 ( t ) , 0 < t ≤ 1 ,

where u 0 ( x ) , u 1 ( t ) , and u 2 ( t ) are the given functions.

4.1 The algorithm of the method

First, let us define the following space function:

ω N = span { C i m ( x ) C j m ( t ) : i , j = 0 , 1 , … , N } .

Then, any function χ N ( x , t ) ∈ ω N may be written as

(15) χ N ( x , t ) = ∑ i = 0 N ∑ j = 0 N v i j C i m ( x ) C j m ( t ) = C m ( x ) T V C m ( t ) ,

where

C m ( t ) = [ C 0 m ( t ) , C 1 m ( t ) , … , C N m ( t ) ] T ,

and the matrix V is given by

V = v 00 v 01 ⋯ v 0 N v 10 v 11 ⋯ w 1 N ⋮ ⋮ ⋱ ⋮ v N 0 v N 1 ⋯ v N N .

Now, Eq. (15) enables us to write the residual Res ( x , t ) of Eq. (14) as

(16) Res ( x , t ) = χ t N ( x , t ) − χ x x N ( x , t ) + α χ N ( x , t ) + ( χ N ( x , t ) ) 3 − ( α + 1 ) ( χ N ( x , t ) ) 2 .

In virtue of Eqs (12), (13), and (15), we can write the residual Res ( x , t ) in (16) in the matrix form as

Res ( x , t ) = C m ( x ) T V ℋ C m ( t ) − ( ℱ C m ( x ) ) T V C m ( t ) + α C m ( x ) T V C m ( t ) + ( C m ( x ) T V C m ( t ) ) 3 − ( α + 1 ) ( C m ( x ) T V C m ( t ) ) 2 .

To obtain the expansion coefficients c i j , we apply the spectral collocation method by forcing the residual R ( x , t ) to be zero at some collocation points ( x i , t j ) as follows:

Res i N + 1 , j N + 1 = 0 , i = 1 , 2 , 3 , … , N − 1 , j = 1 , 2 , 3 , … , N .

Moreover, we obtain the following initial and boundary conditions:

C m i N + 1 T V C m ( 0 ) = u 0 i N + 1 , i = 1 , 2 , 3 , … , N + 1 , C m ( 0 ) T V C m j N + 1 = u 1 j N + 1 , j = 1 , 2 , 3 , … , N , C m ( 1 ) T V C m j N + 1 = u 2 j N + 1 , j = 1 , 2 , 3 , … , N .

Now we have a non-linear system of equations of dimension ( N + 1 ) 2 in the unknown expansion coefficients v i j , which may be solved using Newton’s iterative method.

Remark 3

The elements of the matrix C m ( 0 ) can be obtained from the following relation:

C i m ( 0 ) = ( m ) i 2 i 2 ! , if i even , 0 , if i odd .

5 Convergence and error analysis

This section is interested in analyzing in detail the double expansion in terms of the convolved Fibonacci polynomials for the case corresponding to the positive integer m . Some lemmas and inequalities are required for such analysis.

Lemma 2

The following inequality holds [56]:

∑ i = 0 ∞ x n + 2 i i ! ( i + n ) ! = I n ( 2 x ) ,

where I n ( x ) is the modified Bessel function of order n of the first kind.

Lemma 3

The following inequality holds [57]:

∣ I n ( x ) ∣ ≤ x n cosh ( x ) 2 n Γ ( n + 1 ) , x > 0 .

Lemma 4

Consider the infinitely differentiable function f ( x ) at the origin. We have

f ( x ) = ∑ k = 0 ∞ ∑ j = 0 ∞ ( − 1 ) j f ( 2 j + k ) ( 0 ) j ! ( m ) k ( k + m + 1 ) j C k m ( x ) .

Proof

Suppose that f ( x ) can be expanded as

f ( x ) = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! x n .

As a result of relation (6), the previous equation becomes

f ( x ) = ∑ n = 0 ∞ ∑ r = 0 n 2 ( − 1 ) r f ( n ) ( 0 ) r ! ( m ) n − 2 r ( n + m − 2 r + 1 ) r C n − 2 r m ( x ) ,

which is equivalent to

(17) f ( x ) = ∑ n = 0 ∞ ∑ r = 0 n ( − 1 ) n − r 2 f ( n ) ( 0 ) a n + r n − r 2 ! ( m ) r ( m + r + 1 ) n − r 2 C r m ( x ) .

Now, expanding the right-hand side of Eq. (17) and rearranging the similar terms, the following expansion is obtained

f ( x ) = ∑ k = 0 ∞ ∑ j = 0 ∞ ( − 1 ) j f ( 2 j + k ) ( 0 ) j ! ( m ) k ( k + m + 1 ) j C k m ( x ) .

This completes the proof of this lemma.□

Lemma 5

The following inequality holds for C k m ( x ) :

(18) ∣ C i m ( x ) ∣ ≤ ( 2 m ) i , x ∈ [ 0 , 1 ] .

Proof

We proceed by induction on i . Assume that (18) is valid for ( i − 1 ) and ( i − 2 ) , one obtains

(19) ∣ C i − 1 m ( x ) ∣ ≤ ( 2 m ) i − 1 and ∣ C i − 2 m ( x ) ∣ ≤ ( 2 m ) i − 2 .

The application of recurrence relation (3) along with inequalities (19) enable us to write

∣ C i m ( x ) ∣ = m + i − 1 i x C i − 1 m ( x ) + 2 m + i − 2 i C i − 2 m ( x ) ≤ m ( 2 m ) i − 1 + 2 m ( 2 m ) i − 2 = ( 2 m ) i − 1 ( 1 + m ) .

Thanks to the following identity:

1 + m ≤ 2 m , ∀ m ≥ 1 ,

we obtain the estimation result (18).□

Theorem 2

If f ( x ) is defined on [ 0 , 1 ] and ∣ f ( i ) ( 0 ) ∣ ≤ λ i , i > 0 , where λ is a positive constant and f ( x ) = ∑ k = 0 ∞ c k C k m ( x ) , we obtain

(20) ∣ c k ∣ ≤ λ k cosh ( 2 λ ) Γ ( m ) Γ ( k + m ) .

Moreover, the series converges absolutely.

Proof

Based on Lemma 4 and the assumptions of the theorem, we can write

∣ c k ∣ ≤ ∑ j = 0 ∞ λ 2 j + k j ! ( m ) k ( k + m + 1 ) j = ( k + m ) Γ ( m ) ∑ j = 0 ∞ λ 2 j + k j ! Γ ( j + k + m + 1 ) .

The application of Lemma 2 enables us to write the previous inequality as

∣ c k ∣ ≤ ( k + m ) λ − m Γ ( m ) I k + m ( 2 λ ) .

At the end, the estimation in (20) can be obtained after using Lemma 3.

To prove the second part of the theorem, since

∑ k = 0 ∞ ∣ c k C k m ( x ) ∣ ≤ cosh ( 2 λ ) Γ ( m ) ∑ k = 0 ∞ λ k ( 2 m ) k Γ ( k + m ) .

Using the following inequality Γ ( m ) Γ ( k + m ) ≤ 1 Γ ( k + 1 ) , ∀ m ≥ 1 , k ≥ 0 , we obtain

∑ k = 0 ∞ ∣ c k C k m ( x ) ∣ ≤ cosh ( 2 λ ) ∑ k = 0 ∞ λ k ( 2 m ) k Γ ( k + 1 ) = cosh ( 2 λ ) e 2 m λ ,

so the series converges absolutely.□

Theorem 3

If f ( x ) satisfies the hypothesis of Theorem 2, and e N ( x ) = ∑ k = N + 1 ∞ c k C k m ( x ) , then the following error estimation is satisfied:

∣ e N ( x ) ∣ < cosh ( 2 λ ) e 2 m λ ( 2 m λ ) N + 1 ( N + 1 ) ! .

Proof

The definition of e N ( x ) enables us to write

∣ e N ( x ) ∣ = ∑ k = N + 1 ∞ ∣ c k ∣ ∣ C k m ( x ) ∣ ≤ ∑ k = N + 1 ∞ λ k ( 2 m ) k cosh ( 2 λ ) Γ ( m ) Γ ( k + m ) ≤ cosh ( 2 λ ) ∑ k = N + 1 ∞ ( 2 m λ ) k Γ ( k + 1 ) .

Since

∑ k = N + 1 ∞ ( 2 m λ ) k Γ ( k + 1 ) = e 2 m λ 1 − Γ ( N + 1 , 2 m λ ) N ! < e 2 m λ ( 2 m λ ) N + 1 ( N + 1 ) ! ,

where Γ ( . , . ) denotes upper incomplete gamma functions [58]. Then

□ ∣ e N ( x ) ∣ < cosh ( 2 λ ) e 2 m λ ( 2 m λ ) N + 1 ( N + 1 ) ! .

Theorem 4

If a function χ ( x , t ) = g 1 ( x ) g 2 ( t ) = ∑ i = 0 ∞ ∑ j = 0 ∞ v i j C i m ( x ) C j m ( t ) , with ∣ g 1 ( i ) ( 0 ) ∣ ≤ T 1 i and ∣ g 2 ( i ) ( 0 ) ∣ ≤ T 2 i , where T 1 and T 2 are positive constants, then one has

∣ v i j ∣ ≤ T 1 i T 2 j cosh ( 2 T 1 ) cosh ( 2 T 2 ) ( Γ ( m ) ) 2 Γ ( i + m ) Γ ( j + m ) .

Moreover, the series converges absolutely.

Proof

Applying Lemma 4 and using the assumptions of the theorem χ ( x , t ) = g 1 ( x ) g 2 ( t ) , we can write

v i j = ∑ p = 0 ∞ ∑ q = 0 ∞ ( − 1 ) p + q g 1 ( 2 p + i ) ( 0 ) g 2 ( 2 q + j ) ( 0 ) p ! q ! ( m ) i ( m ) j ( i + m + 1 ) p ( j + m + 1 ) q .

Using the assumption ∣ g 1 ( i ) ( 0 ) ∣ ≤ T 1 i and ∣ g 2 ( i ) ( 0 ) ∣ ≤ T 2 i , one obtains

∣ v i j ∣ ≤ ∑ p = 0 ∞ T 1 2 p + i p ! ( m ) i ( i + m + 1 ) p × ∑ q = 0 ∞ T 2 2 q + j q ! ( m ) j ( j + m + 1 ) q .

Now performing similar steps as in the proof of Theorem 2, we obtain the desired result.□

Theorem 5

If χ ( x , t ) satisfies the hypothesis of Theorem 4, then we have the following upper estimate on the truncation error:

∣ χ ( x , t ) − χ N ( x , t ) ∣ < cosh ( 2 T 1 ) cosh ( 2 T 2 ) e 2 m ( T 1 + T 2 ) ( ( 2 m T 1 ) N + 1 + ( 2 m T 2 ) N + 1 ) ( N + 1 ) ! .

Proof

From definitions of χ ( x , t ) and χ N ( x , t ) , we obtain

∣ χ ( x , t ) − χ N ( x , t ) ∣ = ∑ i = 0 ∞ ∑ j = 0 ∞ v i j C i m ( x ) C j m ( t ) − ∑ i = 0 N ∑ j = 0 N v i j C i m ( x ) C j m ( t ) ≤ ∑ i = 0 N ∑ j = N + 1 ∞ v i j C i m ( x ) C j m ( t ) + ∑ i = N + 1 ∞ ∑ j = 0 ∞ v i j C i m ( x ) C j m ( t ) .

If we used Theorem 4, Lemma 5, and the following inequalities

∑ i = 0 N ( 2 m T 1 ) i i ! = e 2 m T 1 Γ ( N + 1 , 2 m T 1 ) N ! < e 2 m T 1 , ∑ j = N + 1 ∞ ( 2 m T 2 ) j j ! = e 2 m T 2 1 − Γ ( N + 1 , 2 m T 2 ) N ! < e 2 m T 2 ( 2 m T 2 ) N + 1 ( N + 1 ) ! , ∑ j = 0 ∞ ( 2 m T 2 ) j j ! = e 2 m T 2 ,

we obtain the following desired estimation:

∣ χ ( x , t ) − χ N ( x , t ) ∣ < cosh ( 2 T 1 ) cosh ( 2 T 2 ) e 2 m ( T 1 + T 2 ) [ ( 2 m T 1 ) N + 1 + ( 2 m T 2 ) N + 1 ] ( N + 1 ) ! .

This completes the proof of this theorem.□

6 Illustrative examples

In this section, we present three illustrative test problems that will demonstrate the efficiency and applicability of our proposed collocation algorithm. A number of figures and tables will support our numerical results.

Test Problem 1

[52] Consider the F–N equation of the form

χ t = χ x x − χ ( α − χ ) ( 1 − χ ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

χ ( x , 0 ) = α B e α x 2 + A e x 2 α B e α x 2 + A e x 2 + k , 0 < x ≤ 1 , χ ( 0 , t ) = α B e α 2 t 2 + A e t ∕ 2 α B e α 2 t 2 + k e α t + A e t ∕ 2 , 0 < t ≤ 1 , χ ( 1 , t ) = α B e 1 2 α ( α t + 2 ) + A e 1 2 ( t + 2 ) α B e 1 2 α ( α t + 2 ) + k e α t + A e 1 2 ( t + 2 ) , 0 < t ≤ 1 ,

where A , B , k are constants and χ ( x , t ) = A e 1 2 − α t + 2 x 2 + α B e α 2 − 1 α t + 1 2 2 α x A e 1 2 − α t + 2 x 2 + α B e α 2 − 1 α t + 1 2 2 α x + k is the exact solution of this problem.

Table 1 illustrates the maximum absolute error (MAE) at different values of m and α when N = 9 and A = B = k = 1 . Table 2 presents a comparison of MAE between our method at α = 0.7 , N = 9 and the method in the study of Shekarabi et al. [52] when t = 1 . Figure 1 shows the absolute error (AE) at α = 0.2 , A = 3 , B = 2 , k = 4 , m = 4 at different values of N . Table 3 reports the AE at different values of m when N = 9 , α = 1 , A = 3 , B = 2 , k = 4 .

Table 1

MAE of Test Problem 1 at N = 9

x	m	α = 0.1	α = 0.5	α = 0.9
0.1	2	2.19731 × 1 0 ‒ 11	4.50251 × 1 0 ‒ 12	1.23945 × 1 0 ‒ 12
	3	1.03194 × 1 0 ‒ 11	4.92795 × 1 0 ‒ 12	7.81841 × 1 0 ‒ 12
	4	1.07069 × 1 0 ‒ 11	5.18541 × 1 0 ‒ 12	8.27982 × 1 0 ‒ 12
0.2	2	2.04636 × 1 0 ‒ 11	7.6068 × 1 0 ‒ 12	2.08944 × 1 0 ‒ 12
	3	1.64731 × 1 0 ‒ 11	7.80298 × 1 0 ‒ 12	1.35685 × 1 0 ‒ 12
	4	1.63297 × 1 0 ‒ 11	8.06821 × 1 0 ‒ 11	1.40234 × 1 0 ‒ 12
0.3	2	1.94295 × 1 0 ‒ 11	1.08764 × 1 0 ‒ 11	5.0141 × 1 0 ‒ 12
	3	2.27339 × 1 0 ‒ 11	1.08109 × 1 0 ‒ 11	1.95325 × 1 0 ‒ 11
	4	2.20297 × 1 0 ‒ 11	1.10961 × 1 0 ‒ 11	2.00044 × 1 0 ‒ 11
0.4	2	1.80744 × 1 0 ‒ 11	1.41405 × 1 0 ‒ 11	7.47702 × 1 0 ‒ 12
	3	2.85661 × 1 0 ‒ 11	1.37838 × 1 0 ‒ 11	2.56132 × 1 0 ‒ 11
	4	2.72592 × 1 0 ‒ 11	1.41009 × 1 0 ‒ 11	2.61272 × 1 0 ‒ 11
0.5	2	1.62911 × 1 0 ‒ 11	1.74757 × 1 0 ‒ 11	9.40981 × 1 0 ‒ 12
	3	3.41581 × 1 0 ‒ 11	1.68016 × 1 0 ‒ 11	3.18365 × 1 0 ‒ 11
	4	3.21921 × 1 0 ‒ 11	1.71633 × 1 0 ‒ 11	3.24212 × 1 0 ‒ 11
0.6	2	1.45584 × 1 0 ‒ 11	2.11389 × 1 0 ‒ 11	1.06516 × 1 0 ‒ 11
	3	4.03111 × 1 0 ‒ 11	2.01208 × 1 0 ‒ 11	3.81509 × 1 0 ‒ 11
	4	3.76181 × 1 0 ‒ 11	2.05382 × 1 0 ‒ 11	3.88378 × 1 0 ‒ 11
0.7	2	1.43856 × 1 0 ‒ 11	2.56901 × 1 0 ‒ 11	1.09192 × 1 0 ‒ 11
	3	4.88956 × 1 0 ‒ 11	2.43003 × 1 0 ‒ 11	4.43994 × 1 0 ‒ 11
	4	4.53971 × 1 0 ‒ 11	2.47831 × 1 0 ‒ 11	4.52255 × 1 0 ‒ 11
0.8	2	1.94441 × 1 0 ‒ 11	3.23629 × 1 0 ‒ 11	9.92084 × 1 0 ‒ 12
	3	6.39863 × 1 0 ‒ 11	3.05744 × 1 0 ‒ 11	5.04297 × 1 0 ‒ 11
	4	5.95941 × 1 0 ‒ 11	3.11289 × 1 0 ‒ 11	5.14367 × 1 0 ‒ 11
0.9	2	3.05115 × 1 0 ‒ 11	4.07466 × 1 0 ‒ 11	5.36604 × 1 0 ‒ 12
	3	8.68436 × 1 0 ‒ 11	3.85372 × 1 0 ‒ 11	5.32807 × 1 0 ‒ 11
	4	8.14334 × 1 0 ‒ 11	3.91635 × 1 0 ‒ 11	5.45192 × 1 0 ‒ 11

Table 2

Comparison of MAE of Test Problem 1 at t = 1

x	Shekarabi et al. [52]	Our method at α = 0.7 , N = 9
		m = 2	m = 5	m = 8
0.1	1.41366 × 1 0 ‒ 6	1.43141 × 1 0 ‒ 12	5.35527 × 1 0 ‒ 12	5.11991 × 1 0 ‒ 12
0.2	1.39287 × 1 0 ‒ 6	4.82969 × 1 0 ‒ 12	8.46234 × 1 0 ‒ 12	8.21332 × 1 0 ‒ 12
0.3	1.18067 × 1 0 ‒ 6	8.18101 × 1 0 ‒ 12	1.17353 × 1 0 ‒ 11	1.14688 × 1 0 ‒ 12
0.4	1.16488 × 1 0 ‒ 6	1.13429 × 1 0 ‒ 11	1.50149 × 1 0 ‒ 11	1.47282 × 1 0 ‒ 12
0.5	8.85464 × 1 0 ‒ 7	1.43858 × 1 0 ‒ 11	1.83521 × 1 0 ‒ 11	1.80427 × 1 0 ‒ 11
0.6	9.60502 × 1 0 ‒ 7	1.74957 × 1 0 ‒ 11	2.19061 × 1 0 ‒ 11	2.15731 × 1 0 ‒ 11
0.7	5.91894 × 1 0 ‒ 7	2.10611 × 1 0 ‒ 11	2.60294 × 1 0 ‒ 11	2.56745 × 1 0 ‒ 11
0.8	7.75841 × 1 0 ‒ 7	2.59556 × 1 0 ‒ 11	3.15455 × 1 0 ‒ 11	3.11743 × 1 0 ‒ 11
0.9	3.12309 × 1 0 ‒ 7	3.14111 × 1 0 ‒ 11	3.76161 × 1 0 ‒ 11	3.72399 × 1 0 ‒ 11

$Figure 1 AE of Test Problem 1 at α = 0.2 \alpha =0.2 , A = 3 A=3 , B = 2 B=2 , k = 4 k=4 , m = 4 m=4 .$

Figure 1

AE of Test Problem 1 at α = 0.2 , A = 3 , B = 2 , k = 4 , m = 4 .

Table 3

AE of Test Problem 1 at N = 9 , α = 1 , A = 3 , B = 2 , k = 4

x	t	m = 1	m = 2	m = 3
0.2	0.3	2.87548 × 1 0 ‒ 12	2.56258 × 1 0 ‒ 11	2.22342 × 1 0 ‒ 11
	0.6	1.65741 × 1 0 ‒ 10	2.73686 × 1 0 ‒ 11	2.23427 × 1 0 ‒ 11
	0.9	5.61658 × 1 0 ‒ 10	1.67802 × 1 0 ‒ 11	1.10286 × 1 0 ‒ 11
0.4	0.3	7.49513 × 1 0 ‒ 11	4.48604 × 1 0 ‒ 11	4.21652 × 1 0 ‒ 11
	0.6	2.63408 × 1 0 ‒ 10	4.64683 × 1 0 ‒ 11	4.30006 × 1 0 ‒ 11
	0.9	6.99715 × 1 0 ‒ 10	2.55609 × 1 0 ‒ 11	2.42711 × 1 0 ‒ 11
0.6	0.3	1.52773 × 1 0 ‒ 10	6.53161 × 1 0 ‒ 11	6.33881 × 1 0 ‒ 11
	0.6	3.81818 × 1 0 ‒ 10	6.35316 × 1 0 ‒ 11	6.17868 × 1 0 ‒ 11
	0.9	9.02174 × 1 0 ‒ 10	2.94935 × 1 0 ‒ 11	3.33416 × 1 0 ‒ 11
0.8	0.3	2.42303 × 1 0 ‒ 10	8.66958 × 1 0 ‒ 11	8.58721 × 1 0 ‒ 11
	0.6	5.32292 × 1 0 ‒ 10	7.70678 × 1 0 ‒ 11	7.76854 × 1 0 ‒ 11
	0.9	1.19363 × 1 0 ‒ 9	2.55119 × 1 0 ‒ 11	3.61877 × 1 0 ‒ 11

Test Problem 2

[59] Consider the F–N equation of the form

χ t = χ x x − χ ( α − χ ) ( 1 − χ ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

χ ( x , 0 ) = e − x 2 − e − x 2 − 2 + 1 , 0 < x ≤ 1 , χ ( 0 , t ) = 2 e t 2 + 2 , 0 < t ≤ 1 , χ ( 1 , t ) = 2 e 1 2 e t 2 + 2 e 1 2 , 0 < t ≤ 1 ,

where χ ( x , t ) = e t 2 − x 2 − e t 2 − x 2 − 2 + 1 is the exact solution of this problem when α = 1 .

Table 4 illustrates the MAE at different values of m when N = 9 . Figure 2 shows the AE at different values of N when m = 9 . Table 5 presents a comparison of MAE between our method at N = 9 , m = 5 and the method of Yokus [59] when t = 0.001 .

Table 4

MAE of Test Problem 2 at N = 9

x	m = 1	m = 2	m = 3	m = 4
0.1	4.92329 × 1 0 ‒ 11	1.42201 × 1 0 ‒ 11	1.24963 × 1 0 ‒ 11	7.94825 × 1 0 ‒ 11
0.2	4.06755 × 1 0 ‒ 11	2.22037 × 1 0 ‒ 11	2.12791 × 1 0 ‒ 11	1.73394 × 1 0 ‒ 11
0.3	3.35761 × 1 0 ‒ 11	3.05615 × 1 0 ‒ 11	3.03421 × 1 0 ‒ 11	2.69274 × 1 0 ‒ 11
0.4	2.79586 × 1 0 ‒ 11	3.92781 × 1 0 ‒ 11	3.96664 × 1 0 ‒ 11	3.67318 × 1 0 ‒ 11
0.5	2.38461 × 1 0 ‒ 11	4.83803 × 1 0 ‒ 11	4.92889 × 1 0 ‒ 11	4.68149 × 1 0 ‒ 11
0.6	2.15862 × 1 0 ‒ 11	5.76201 × 1 0 ‒ 11	5.89821 × 1 0 ‒ 11	5.69655 × 1 0 ‒ 11
0.7	2.19767 × 1 0 ‒ 11	6.63495 × 1 0 ‒ 11	6.81311 × 1 0 ‒ 11	6.65719 × 1 0 ‒ 11
0.8	2.62933 × 1 0 ‒ 11	7.34869 × 1 0 ‒ 11	7.56941 × 1 0 ‒ 11	7.45911 × 1 0 ‒ 11
0.9	4.00921 × 1 0 ‒ 11	7.36962 × 1 0 ‒ 11	7.64045 × 1 0 ‒ 11	7.57291 × 1 0 ‒ 11

Figure 2

AE of Test Problem 2 at m = 9 .

Table 5

Comparison of AE of Test Problem 2 at t = 0.001

x	Yokus [59]	Our method at N = 9 , m = 5
0.00	9.260091360374645 × 1 0 ‒ 9	1.53093 × 1 0 ‒ 11
0.001	9.266945988350983 × 1 0 ‒ 9	1.48731 × 1 0 ‒ 11
0.002	9.273449896873842 × 1 0 ‒ 9	1.44468 × 1 0 ‒ 11
0.003	9.279814694451716 × 1 0 ‒ 9	1.40307 × 1 0 ‒ 11
0.004	9.286474367264930 × 1 0 ‒ 9	1.36243 × 1 0 ‒ 11
0.005	9.292864144860857 × 1 0 ‒ 9	1.32275 × 1 0 ‒ 11
0.006	9.299528924699985 × 1 0 ‒ 9	1.28401 × 1 0 ‒ 11

Test Problem 3

[50] Consider the F–N equation of the form

χ t = χ x x − χ ( α − χ ) ( 1 − χ ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

χ ( x , 0 ) = 1 2 tanh x 2 2 + 1 2 , 0 < x ≤ 1 , χ ( 0 , t ) = 1 2 − 1 2 tanh 1 4 ( 2 α − 1 ) t , 0 < t ≤ 1 , χ ( 1 , t ) = 1 2 tanh 1 − ( 2 α − 1 ) t 2 2 2 + 1 2 , 0 < t ≤ 1 ,

where χ ( x , t ) = 1 2 tanh x − ( 2 α − 1 ) t 2 2 2 + 1 2 is the exact solution of this problem.

Table 6 reports the MAE at different values of m and α when N = 9 . Figure 3 shows the AE at different values of N when α = 0.6 and m = 9 . Figure 4 illustrates the AE at different values of N when α = 1 and m = 10 .

Table 6

MAE of Test Problem 3 at N = 9

x	m	α = 0.2	α = 0.5	α = 0.8
0.1	2	8.22231 × 1 0 ‒ 13	3.90104 × 1 0 ‒ 11	3.53932 × 1 0 ‒ 12
	3	1.33494 × 1 0 ‒ 11	8.19234 × 1 0 ‒ 13	4.97791 × 1 0 ‒ 12
	4	1.76525 × 1 0 ‒ 11	1.59109 × 1 0 ‒ 11	4.78867 × 1 0 ‒ 12
0.2	2	6.61105 × 1 0 ‒ 12	3.09197 × 1 0 ‒ 11	4.03312 × 1 0 ‒ 11
	3	1.95906 × 1 0 ‒ 11	1.02171 × 1 0 ‒ 11	4.34802 × 1 0 ‒ 12
	4	2.37476 × 1 0 ‒ 11	2.47622 × 1 0 ‒ 11	7.34085 × 1 0 ‒ 12
0.3	2	1.28556 × 1 0 ‒ 11	2.45207 × 1 0 ‒ 11	4.63161 × 1 0 ‒ 11
	3	2.60576 × 1 0 ‒ 11	2.09476 × 1 0 ‒ 11	3.54351 × 1 0 ‒ 12
	4	3.01908 × 1 0 ‒ 11	3.40069 × 1 0 ‒ 11	9.86106 × 1 0 ‒ 12
0.4	2	1.79599 × 1 0 ‒ 11	2.05599 × 1 0 ‒ 11	5.2865 × 1 0 ‒ 11
	3	3.20018 × 1 0 ‒ 11	3.12795 × 1 0 ‒ 11	3.16708 × 1 0 ‒ 12
	4	3.62178 × 1 0 ‒ 11	4.29387 × 1 0 ‒ 11	1.17295 × 1 0 ‒ 11
0.5	2	2.21297 × 1 0 ‒ 11	1.87729 × 1 0 ‒ 11	6.02974 × 1 0 ‒ 11
	3	3.76819 × 1 0 ‒ 11	4.13879 × 1 0 ‒ 11	3.02347 × 1 0 ‒ 12
	4	4.20808 × 1 0 ‒ 11	5.18812 × 1 0 ‒ 11	1.31209 × 1 0 ‒ 11
0.6	2	2.64101 × 1 0 ‒ 11	1.80586 × 1 0 ‒ 11	6.9836 × 1 0 ‒ 11
	3	4.4207 × 1 0 ‒ 11	5.22464 × 1 0 ‒ 11	2.03781 × 1 0 ‒ 12
	4	4.88888 × 1 0 ‒ 11	6.20053 × 1 0 ‒ 11	1.50911 × 1 0 ‒ 11
0.7	2	3.32897 × 1 0 ‒ 11	1.58654 × 1 0 ‒ 11	8.41951 × 1 0 ‒ 11
	3	5.41502 × 1 0 ‒ 11	6.62178 × 1 0 ‒ 11	2.39697 × 1 0 ‒ 12
	4	5.92226 × 1 0 ‒ 11	7.59359 × 1 0 ‒ 11	2.01715 × 1 0 ‒ 11
0.8	2	4.82165 × 1 0 ‒ 11	6.58584 × 1 0 ‒ 12	1.08967 × 1 0 ‒ 10
	3	7.30509 × 1 0 ‒ 11	8.86379 × 1 0 ‒ 11	1.54486 × 1 0 ‒ 11
	4	7.86533 × 1 0 ‒ 11	9.9328 × 1 0 ‒ 11	3.37141 × 1 0 ‒ 11
0.9	2	7.28945 × 1 0 ‒ 11	1.02841 × 1 0 ‒ 11	1.48358 × 1 0 ‒ 10
	3	1.02785 × 1 0 ‒ 10	1.19659 × 1 0 ‒ 10	4.13357 × 1 0 ‒ 11
	4	1.09067 × 1 0 ‒ 10	1.32765 × 1 0 ‒ 10	5.97471 × 1 0 ‒ 11

$Figure 3 AE of Test Problem 3 at α = 0.6 , m = 9 \alpha =0.6,m=9 .$

Figure 3

AE of Test Problem 3 at α = 0.6 , m = 9 .

$Figure 4 AE of Test Problem 3 at α = 1 , m = 10 \alpha =1,m=10 .$

Figure 4

AE of Test Problem 3 at α = 1 , m = 10 .

Remark 4

Some of the tables presented show that the approximations for the case correspond to m = 1 , which means that the Fibonacci case is not always the best among the other approximations. This, of course, demonstrates the importance of the parameter involved in the convolved Fibonacci polynomials.

7 Concluding remarks

In this article, we presented a spectral collocation algorithm for the numerical treatment of the F–N equation. The core of the derivation of the presented algorithm was the employment of the operational matrices of derivatives of the convolved Fibonacci polynomials. These operational matrices were established using new derivative expressions of the convolved Fibonacci polynomials. Other new formulas and inequalities regarding the convolved Fibonacci polynomials were established to be able to discuss in detail the convergence and the truncation error of the double expansion of the proposed approximate solutions. Some numerical experiments were given to show the performance of the proposed algorithm. As far as we know, the used expansion for obtaining the approximate solution is new. We hope in the near future to use the same expansion to treat other important differential equations. In addition, we hope to introduce other generalizations and modifications of Fibonacci polynomials and employ them in numerical analysis and approximation theory. All codes were written and debugged by Mathematica 11 on HP Z420 Workstation, Processor: Intel (R) Xeon(R) CPU E5-1620 - 3.6 GHz, 16GB Ram DDR3, and 512 GB storage.

Acknowledgments

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-23-FR-42). Therefore, the authors thank the University of Jeddah for its technical and financial support.

Funding information: The article was funded by the University of Jeddah, Jeddah, Saudi Arabia.
Author contributions: All authors accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: The authors did not use any scientific data during this research.

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Received: 2023-08-06

Revised: 2023-09-11

Accepted: 2023-09-12

Published Online: 2024-08-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0332

Keywords for this article

Fitzhugh–Nagumo equation; convolved Fibonacci polynomials; collocation method; operational matrices; convergence analysis

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