Machine learning approach of Casson hybrid nanofluid flow over a heated stretching surface

1.   Introduction

Fractional calculus is a subdivision of classical calculus that concerns itself with derivatives as well as integrals of nearly any fractional order. This mathematical field has gained significant attention due to its ability to model complex phenomena more accurately than integer-order calculus. Fractional calculus offers robust methods for characterizing memory and hereditary features of different materials and processes, rendering it essential in disciplines such as control theory ^[14], signal processing ^[4], and differential equations ^[11,12,13]. The foundations of fractional calculus were laid by early mathematicians such as Leibniz, Liouville, and Riemann, who explored the concept of generalizing the order of differentiation and integration beyond integers ^[3,7]. Modern advancements have further developed these ideas, leading to a robust theoretical framework and numerous practical applications.

The nabla difference operator is a discrete counterpart of the continuous derivative, employed in discrete calculus, specifically in the context of time-scale calculus. This operator, often denoted by $\nabla$, operates on functions defined on discrete domains, such as sequences or time scales. The nabla difference operator plays a crucial role in various fields, including exact analysis, discrete dynamic systems, and operational calculus in ^[9,21,22]. It facilitates the formulation and solution of difference equations, which are the discrete counterparts of differential equations. The papers "On the definitions of nabla fractional operators" and "Discrete fractional calculus includes the nabla operator" go into great detail about what nabla fractional operators are and how they can be used in real life. They stress how important they are in the field of discrete mathematics ^[1,2]. Recently, the nabla difference operator has been seen in sequential differences in the nabla fractional calculus, the combined delta-nabla sum operator in discrete fractional calculus, and discrete fractional calculus consisting of the nabla operator in ^[2,20]. These studies contribute to a deeper understanding of the interplay between discrete and continuous analysis. In research ^[15], such as on K$\ddot{o}$the-Toeplitz duals of generalized difference sequence spaces and Laplace transforms for the nabla-difference operator, investigations into the theoretical underpinnings and applications of these operators in discrete calculus ^[6].

The Fibonacci sequence is a widely recognized integer sequence defined by the recurrence relation ${\tt{F}}_{r} = {\tt{F}}_{r-1} + {\tt{F}}_{r-2}$, where ${\tt{F}}_{0}$ is $0$ and ${\tt{F}}_{1}$ is $1$. This sequence appears in various natural phenomena and has numerous applications in computer science, mathematics, and financial modeling. The study of sequence spaces derived from difference operators has gained significant attention, particularly with generalized Fibonacci difference operators. The derivation of sequence spaces arising from triple-band generalized Fibonacci difference operator ^[18,24], investigates the structural properties of these spaces, while generalized Fibonacci difference sequence spaces and compact operators ^[17,23] explores their impact on the boundedness and compactness of sequences. These works extend classical sequence space theory by integrating the recursive properties of Fibonacci sequences, highlighting new connections in functional analysis.

The motivation for this study arises from the need to further explore and expand the theoretical systems and applications of the nabla difference operator in generating and analyzing Fibonacci sequences. We present the trigonometric nabla difference operator of order 2 and its discrete integral, analyzing the $\overline{\theta({\tt{t}})}$-sequences, their summation, and the proportional derivative of the $\overline{\theta({\tt{t}})}$-polynomials.

The contributions of the research are delineated as follows:

1. By utilizing various trigonometric coefficients in the $\overline{\theta({\tt{t}})}$-Fibonacci equation and its inverses, we formulate new sequences and analyze their characteristics.

2. The nabla difference operator of order 2 facilitates the generation of $\overline{\theta({\tt{t}})}$-Fibonacci sequences and their exact and numerical solutions.

3. We provide chaotic behavior of the generating function of the second-order Fibonacci sequence with the coefficients of trigonometric functions.

4. The study includes MATLAB examples to demonstrate the practical applications of our theoretical findings.

Throughout this article, we make use of the notations and elucidations from the following Table 1.

2.   Second-order $\overline{\theta({\tt{t}})}$-Fibonacci sequence and its sum

Here, we formulate a generic nabla-difference operator that includes a trigonometric co-efficient $\mathop \nabla \limits_{\overline{\theta({\tt{t}})}}v({\tt{t}}) = v({\tt{t}})-{\alpha}_{1}sin(b_1{\tt{t}}) v({\tt{t}}_{0, 1})-{\alpha}_{2}sin(b_2{\tt{t}}) v({\tt{t}}_{0, 2})$ which generates second-order $\overline{\theta({\tt{t}})}$-Fibonacci sequence and its sum.

Definition 2.1. Let ${\tt{t}}$ be any positive real number and $n\ge 2$; a second-order generic $\overline{\theta({\tt{t}})}$-sequence is defined recurrently as $\mathcal{F}_{{\tt{t}}, 0} = 1, \ \ \mathcal{F}_{{\tt{t}}, 1} = \alpha_1sin(b_1{\tt{t}})$, and

Definition 2.2. For any positive real number ${\tt{t}}$, a generic nabla difference operator of order two using sine (any trigonometric) function coefficients on $v({\tt{t}})$, denoted as $\mathop \nabla \limits_{\overline{\theta({\tt{t}})}}v({\tt{t}})$, is defined as

and the inverse of $\mathop\nabla\limits_{\overline{\theta({\tt{t}})}}v({\tt{t}}) = u({\tt{t}})$ is defined as

Definition 2.3. ^[16] A proportional ${\tt{\alpha}}$- derivative is defined by

is the proportional $\alpha$-derivative of ${\tt{f}}({\tt{t}})$. Here ${\tt{D}}^{\tt{\alpha}}$ is proportional $\alpha$- provided the function ${\tt{f}}({\tt{t}})$ is differentiable at ${\tt{t}}$.

Lemma 2.1. For any real number ${\tt{t}}, v({\tt{t}})$ is a function. Then it leads

Proof: By doing $v({\tt{t}}) = a^{{\tt{t}}_{0, 0}}$ in (2.2), we observe that

Now, the function $v({\tt{t}})$ follows from the Definition 2.2.

Remark 2.1. When $\alpha_1 = 1 = \alpha_2$ in lemma 2.1, then we observe that

Corollary 2.1. For any real number ${\tt{t}}$, $e^{s{\tt{t}}}$ is a function. Then we observe

Proof: By doing $a = e$ in (2.5), we conclude the proof.

Corollary 2.2. For any real number ${\tt{t}}$, {$e^{-s{\tt{t}}}$} is a function. Then we observe

Proof: By doing $a = e^{-1}$ in (2.5), we conclude the proof.

Corollary 2.3. For any real number ${\tt{t}}$, {$e^{-s{\tt{t}}}$} is a function. Then we observe

Proof: By doing $\alpha_1 = \alpha_2 = 1$ in (2.8), we conclude the proof.

Propsition 2.1. If the function $v({\tt{t}}) = \mathop\nabla\limits_{\overline{\theta({\tt{t}})}}^{-1}u({\tt{t}})$ is a $E^*$-solution of (2.2), $\mathcal{F}_{{\tt{t}}, 0} = 1, \mathcal{F}_{{\tt{t}}, 1} = \alpha_{1}sin(b_1{\tt{t}})$ and

$\mathcal{F}_{{\tt{t}}, n+1} = \alpha_1 sin(b_1{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n}+\alpha_2 sin(b_2{\tt{t}}_{n, -1}) \mathcal{F}_{{\tt{t}}, n-1},$ for $i = 0, 1, 2, ...$ then, the $E^*$-solution is equal to a $N^*$-solution, which is given by

Proof: For the function $v({\tt{t}})$, the Definition 2.2 yields

By changing $k$ by $k_{0, 1}$ and then put the value of $v({\tt{t}}_{0, 1})$ into (2.11), we get

$v({\tt{t}}) = u({\tt{t}})+ \mathcal{F}_{{\tt{t}}, 1}u({\tt{t}}_{0, 1})+[\mathcal{F}_{{\tt{t}}, 1}\alpha_1 sin(b_1{\tt{t}}_{0, 1})+$

which leads to $v({\tt{t}}) = \mathcal{F}_{{\tt{t}}, 0}u({\tt{t}})+ \mathcal{F}_{{\tt{t}}, 1}u({\tt{t}}_{0, 1})+$

where $\mathcal{F}_{{\tt{t}}, 0} = 0$, $\mathcal{F}_{{\tt{t}}, 1} = \alpha_{1}sin(b_1{\tt{t}})$ and $\mathcal{F}_{{\tt{t}}, 2} = \mathcal{F}_{{\tt{t}}, n+1} = \alpha_1 sin(b_1{\tt{t}}_{1, 0}) \mathcal{F}_{{\tt{t}}, 1}+\alpha_2 sin(b_2{\tt{t}}_{1, -1}) \mathcal{F}_{{\tt{t}}, 0}.$ By changing $k$ by $k_{0, 2}$ in (2.11) and then putting the value of $v({\tt{t}}_{0, 2})$ into (2.13), we observe

where $\mathcal{F}_{{\tt{t}}, 3} = \mathcal{F}_{{\tt{t}}, 3} = \alpha_1 sin(b_1{\tt{t}}_{2, 0}) \mathcal{F}_{{\tt{t}}, 2}+\alpha_2 sin(b_2{\tt{t}}_{2, -1}) \mathcal{F}_{{\tt{t}}, 1}.$

By carrying out this procedure repeatedly (2.10).

Corollary 2.4. If the function $\mathop\nabla\limits_{\overline{\theta({\tt{t}})}}^{-1}u({\tt{t}}) = v({\tt{t}})$, $\mathcal{F}_{{\tt{t}}, 0} = 1, \mathcal{F}_{{\tt{t}}, 1} = sin(b_1{\tt{t}})$ and

$\mathcal{F}_{{\tt{t}}, n+1} = sin(b_1{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n}+sin(b_2{\tt{t}}_{n, 1}) \mathcal{F}_{{\tt{t}}, n-1},$ for $i = 0, 1, 2, ...$ then

Proof: The proof follows by doing $\alpha_1 = 1 = \alpha_2$ in Theorem 2.1.

Corollary 2.5. Let $v({\tt{t}})$ be any function and

then we observe that

$a^{s{\tt{t}}}- \mathcal{F}_{{\tt{t}}, n+1} a^{s({\tt{t}}_{n, -1})}-\alpha_{2} sin(b_2{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n} a^{s({\tt{t}}_{n, -2})}$

Proof: By doing $v({\tt{t}}) = a^{s{\tt{t}}}$ and applying (2.5) in (2.10), we observed the proof.

The following illustration proves the significance of corollary 2.5.

Corollary 2.6. Let $\alpha_1 = \alpha_2 = 1$ in (2.15). Then we have

$a^{{\tt{t}}_{0, 0}}- \mathcal{F}_{{\tt{t}}, n+1} a^{{\tt{t}}_{n, -1}}-sin(b_2{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n} a^{{\tt{t}}_{n, -2}}$

Proof: By doing $v({\tt{t}}) = a^{{\tt{t}}_{0, 0}}$ and putting (2.5) in (2.10), we complete the proof.

The following illustration proves the significance of corollary 2.6.

Corollary 2.7. Let $e^{s{\tt{t}}}$ be a function of ${\tt{t}}\in(-\infty, \infty)$. Then

$e^{s{\tt{t}}}- \mathcal{F}_{{\tt{t}}, n+1} e^{s({\tt{t}}_{n, -1})}-\alpha_{2} sin(b_2{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n} e^{s({\tt{t}}_{n, -2})}$

Proof: By doing $v({\tt{t}}) = e^{s{\tt{t}}}$ and applying (2.8) in (2.5), we obtain (2.17).

Propsition 2.2. Let $x\in \mathbb N(0)$. Then, the $E^*$-solution of $\overline{\theta({\tt{t}})}$-equation

$v({\tt{t}})-\alpha_1sin(b_1{\tt{t}})v({\tt{t}}_{0, 1})-\alpha_2sin(b_2{\tt{t}})v({\tt{t}}_{0, 2}) = \left[{\tt{t}}^x_{0, 0}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^x_{0, 1}-\alpha_2sin(b_2{\tt{t}}){\tt{t}}^x_{0, 2}\right]$ is

Proof: By doing $v({\tt{t}}) = {\tt{t}}^x_{0, 0}$ with the equation (2.2) and using (2.3), we obtain (2.18).

Corollary 2.8. By doing $x = 2$ with the Theorem 2.2, we observed

which is the $E^*$-solution of the nabla difference equation of order two

$\mathop{\nabla}\limits_{\overline{\theta({\tt{t}})}} v({\tt{t}}) = {\tt{t}}^2_{0, 0}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^2_{0, 1}-\alpha_2sin(b_2{\tt{t}}){\tt{t}}^2_{0, 2}$.

Proof: From (2.18), replacing $x = 2$, we obtain (2.19).

Corollary 2.9. If $v({\tt{t}}) = \mathop{\nabla}\limits_{\overline{\theta({\tt{t}})}}^{-1}\left[{\tt{t}}^x_{0, 0}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^x_{0, 1}-\alpha_2sin(b_2{\tt{t}}){\tt{t}}^x_{0, 2}\right]$ is a $E^*$-solution of the equation (2.18), then

$v({\tt{t}})- \mathcal{F}_{{\tt{t}}, n+1}v({\tt{t}}_{0, -1})-\alpha_{2}sin(b_2{\tt{t}}_{n, 0}) \mathcal{F}_{{\tt{t}}, n} v({\tt{t}}_{0, -2})$

Proof: Taking $u({\tt{t}}) = {\tt{t}}^x_{0, 0}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^x_{0, 1}-\alpha_2sin(b_2{\tt{t}}){\tt{t}}^x_{0, 2}$ in (2.10), we have (2.20).

Propsition 2.3. If $v({\tt{t}})$ is a $E^*$-solution of $sine$-coefficients $\overline{\theta({\tt{t}})}$-equation

$v({\tt{t}})-\alpha_1sin(b_1{\tt{t}})v({\tt{t}}_{0, 1})-\alpha_2sin(b_2{\tt{t}})v({\tt{t}}_{0, 2}) = {\tt{t}}^x_{0, 0}a^{{\tt{t}}_{0, 0}}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^x_{0, 1}a^{{\tt{t}}_{0, 1}}- \alpha_2sin(b_2{\tt{t}}){\tt{t}}^x_{0, 2}a^{{\tt{t}}_{0, 2}}$,

then we have

$v({\tt{t}})- \mathcal{F}_{{\tt{t}}, n+1}({\tt{t}}_{n, -1})-\alpha_2sin(b_2{\tt{t}}_{n, 0})v({\tt{t}}_{n, -2})$

Proof: By doing $u({\tt{t}}) = \left[{\tt{t}}^x_{0, 0}a^{{\tt{t}}_{0, 0}}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^x_{0, 1}a^{{\tt{t}}_{0, 1}}- \alpha_2sin(b_2{\tt{t}}){\tt{t}}^x_{0, 2}a^{{\tt{t}}_{0, 2}}\right]$ in (2.10) and using (2.5), we get (2.21).

Corollary 2.10. A $E^*$-solution of $\overline{\theta({\tt{t}})}$-equation is

and hence we have

${\tt{t}}^3_{0, 0}a^{{\tt{t}}_{0, 0}}- \mathcal{F}_{{\tt{t}}, n+1}{\tt{t}}^3_{n, -1} a^{{\tt{t}}_{n, -1}}-\alpha_2sin(b_2{\tt{t}}_{n, 0}){\tt{t}}^3_{n, -2} a^{{\tt{t}}_{n, -2}}$

Proof: The proof is completed by doing $x = 3$ in Theorem 2.3.

Propsition 2.4. Let $v({\tt{t}})$ be a solution of $\overline{\theta({\tt{t}})}$-equation

$v({\tt{t}})-\alpha_1sin(b_1{\tt{t}})v({\tt{t}}_{0, 1})-\alpha_2sin(b_2{\tt{t}})v({\tt{t}}_{0, 2}) = {\tt{t}}^{(x)}a^{{\tt{t}}_{0, 0}}-\alpha_1sin(b_1{\tt{t}}){\tt{t}}^{(x)}_{0, 1}a^{{\tt{t}}_{0, 1}}-\alpha_2sin(b_2{\tt{t}}){\tt{t}}^{(x)}_{0, 2}a^{{\tt{t}}_{0, 2}}$, then we have

$v({\tt{t}})- \mathcal{F}_{{\tt{t}}, n+1}v({\tt{t}}_{n, -1})-\alpha_2sin(b_2{\tt{t}}_{n, 0})v({\tt{t}}_{n, -2})$

Proof: By doing $v({\tt{t}}) = {\tt{t}}^{(x)}_{0, 0}a^{{\tt{t}}_{0, 0}}$ in (2.10) and using (2.5), we get (2.23).

Corollary 2.11. Let $v({\tt{t}})$ be a solution of $\overline{\theta({\tt{t}})}$-equation

$v({\tt{t}})-sin(b_1{\tt{t}})v({\tt{t}}_{0, 1})-sin(b_2{\tt{t}})v({\tt{t}}_{0, 2}) = {\tt{t}}^{(x)}a^{{\tt{t}}_{0, 0}}-sin(b_1{\tt{t}}){\tt{t}}^{(x)}_{0, 1}a^{{\tt{t}}_{0, 1}}-sin(b_2{\tt{t}}){\tt{t}}^{(x)}_{0, 2}a^{{\tt{t}}_{0, 2}}$, then we have

$v({\tt{t}})- \mathcal{F}_{{\tt{t}}, n+1}v({\tt{t}}_{n, -1})-sin(b_2{\tt{t}}_{n, 0})v({\tt{t}}_{n, -2})$

Proof: By doing $\alpha_1 = \alpha_2 = 1$ in (2.23), we obtain (2.24).

3.   Illustrative numerical examples

The objective of this section is to demonstrate the efficacy of the primary findings presented in this paper by employing precise examples from the literature. Also, we have investigated graphical representations of Fibonacci sequences with the co-efficients of the Sine, Cosine and Co-secant functions for different values of ${\tt{t}}$ in the following Figures 1–3.

The validity of the definition 2.1 is confirmed by the subsequent illustrative example 3.1.

Example 3.1. (i) By doing ${\tt{t}} = 10$, $b_1 = 2$, $b_2 = 1$, $\xi = 0.4$, $\alpha_{1} = 2$, and $\alpha_{2} = 1$ in (2.1), we obtain a Sine-Fibonacci sequence $\{1, 1.8259, 0.7097, -0.9351, 1.9327, -3.9778, \cdots\}$.

(ii) When ${\tt{t}} = 15$, $\xi = 0.5$, $\alpha_1 = 0.2$, $\alpha_2 = 0.1$, $r1 = 3$, and $r2 = 1$ in (2.1), we have a Cosine-Fibonacci sequence $\{1.0000, -1.5194, 1.9182, -3.9008, 3.4203, 0.8856, 5.1684, \cdots\}.$

Also, the sine Fibonacci polynomials are observed by

$\mathcal{F}_0({\tt{t}}) = 1,$

$\mathcal{F}_1({\tt{t}}) = 2sin(2{\tt{t}}),$

$\mathcal{F}_2({\tt{t}}) = 4sin(2{\tt{t}})sin(2{\tt{t}}-0.8)+sin({\tt{t}}),$

$\mathcal{F}_3({\tt{t}}) = 8sin(2{\tt{t}})sin(2{\tt{t}}-1.6)sin(2{\tt{t}}-0.8)+2sin({\tt{t}})sin(2{\tt{t}}-1.6)+2sin({\tt{t}}-0.4)sin(2{\tt{t}})$, etc.

Furthermore, using (2.4), we have the following proportional $\alpha$-derivative of the sine Fibonacci polynomials;

$\mathcal{F}_0^{\alpha}({\tt{t}}) = 0,$

$\mathcal{F}_1^{\alpha}({\tt{t}}) = 4\alpha cos(2{\tt{t}})+2(1-\alpha)sin(2{\tt{t}}),$

$\mathcal{F}_2^{\alpha}({\tt{t}}) = 8\alpha[sin(2{\tt{t}}-0.8)cos2{\tt{t}}+cos(2{\tt{t}}-0.8)sin2{\tt{t}}+cos{\tt{t}}]$

$\qquad \qquad\qquad \qquad\qquad \qquad \qquad\qquad \qquad +(1-\alpha)[4sin(2{\tt{t}}-0.8)sin2{\tt{t}}+sin{\tt{t}}]$,

$\mathcal{F}_3^{\alpha}({\tt{t}}) = \alpha[16cos(2{\tt{t}}-1.6)sin(2{\tt{t}}-0.8)sin(2{\tt{t}})+16sin(2{\tt{t}}-1.6)cos(2{\tt{t}}-0.8)sin(2{\tt{t}})+$

One can derive second-order Fibonacci polynomials and sequences for each pair $\overline{\theta({\tt{t}})}\in \mathbb R^2$. The validity of the result 2.5 is confirmed by the subsequent illustrative example 3.2.

Example 3.2. Setting ${\tt{t}} = 11$, $b_1 = 2$, $\alpha_{1} = 3$, $s = 2$, $\alpha_{2} = 2$, $n = 2$, $\xi = 0.3$, $b_2 = 1$, and $a = 3$ in $(2.15)$, we obtain the following:

(i) The sum of the sine-Fibonacci series

where $\mathcal{F}_{{\tt{t}}, 0} = 1$, $\mathcal{F}_{{\tt{t}}, 1} = -0.03$, $\mathcal{F}_{{\tt{t}}, 2} = -2.04$, $\mathcal{F}_{{\tt{t}}, 3} = -5.65$.

(ii) The sum of the cosine-Fibonacci series

where $\mathcal{F}_{{\tt{t}}, 0} = 1$, $\mathcal{F}_{{\tt{t}}, 1} = -3.00$, $\mathcal{F}_{{\tt{t}}, 2} = 7.48, \mathcal{F}_{{\tt{t}}, 3} = -6.57$,

and the cosine Fibonacci polynomials are given by

$\mathcal{F}_0({\tt{t}}) = 1$,

$\mathcal{F}_1({\tt{t}}) = 3cos(2{\tt{t}}),$

$\mathcal{F}_2({\tt{t}}) = 9cos(2{\tt{t}}-0.6)cos(2{\tt{t}})+2cos({\tt{t}})$

$\mathcal{F}_3({\tt{t}}) = 27cos(2{\tt{t}}-1.2)cos(2{\tt{t}}-0.6)cos(2{\tt{t}})+6cos(2{\tt{t}}-1.2)cos({\tt{t}})+2cos({\tt{t}}-0.4)cos(2{\tt{t}})$, etc. Furthermore, using (2.4), we have the following proportional $\alpha$-derivative of the co-secant Fibonacci polynomials;

$\mathcal{F}_0^{\alpha}({\tt{t}}) = 0, \mathcal{F}_1^{\alpha}({\tt{t}}) = \alpha[-6sin(2{\tt{t}})]+3(1-\alpha)cos(2{\tt{t}}),$

$\mathcal{F}_2^{\alpha}{\tt{t}}) = \alpha[-18sin(2{\tt{t}}-0.6)cos(2{\tt{t}})-18cos(2{\tt{t}}-0.6)$

$sin(2{\tt{t}})-2sin{\tt{t}}]+(1-\alpha)9cos(2{\tt{t}}-0.6)cos(2{\tt{t}})+2cos({\tt{t}})],$ and etc.

The validity of the definition 2.7 is confirmed by the subsequent illustrative example 3.3.

Example 3.3. By doing $\alpha_1 = 2$, ${\tt{t}} = 10$, $\alpha_2 = 1$, $\xi = 0.4$, $b_1 = 2$, $n = 2$, and $b_2 = 1$ in $(2.16)$, we have the following:

(i) The sine-Fibonacci sum of the series is

$e^{-9}- F_4e^{5}-(0.3) 6^2 \mathcal{F}_{{\tt{t}}, 3} e^{-4} = \sum\limits_{i = 0}^{3} \mathcal{F}_{{\tt{t}}, i} e^{-(9-i)}[1-(0.8)(9-i)^3e-(0.3)(9-i)^2e^2]$

$= 523194317.45,$ where $\mathcal{F}_{{\tt{t}}, 0} = 1$, $\mathcal{F}_{{\tt{t}}, 1} = 1.83$, $\mathcal{F}_{{\tt{t}}, 2} = 0.71$, and $\mathcal{F}_{{\tt{t}}, 3} = -0.23$.

(ii) The co-secant-Fibonacci sum of the series is

$e^{-9}- F_4e^{5}-(0.3) 6^2 \mathcal{F}_{{\tt{t}}, 3} e^{-4} = \sum\limits_{i = 0}^{3} \mathcal{F}_{{\tt{t}}, i} e^{-(9-i)} \left[1-(0.8)(9-i)^3e-(0.3)(9-i)^2e^2\right] = 1.49,$ where $\mathcal{F}_{{\tt{t}}, 0} = 1$, $\mathcal{F}_{{\tt{t}}, 1} = 5.31$, $\mathcal{F}_{{\tt{t}}, 2} = 0.71$ and $\mathcal{F}_{{\tt{t}}, 3} = -0.94$.

Also, taking $\alpha_1 = 1$, $b_1 = 1$, $\xi = 0.5$, $\alpha_2 = 2$, and $b_3 = 1$ in $(2.16)$, we have the following co-secant Fibonacci polynomials:

$\mathcal{F}_0({\tt{t}}) = 1, \ \ \mathcal{F}_1({\tt{t}}) = cosec(2t), \ \ \mathcal{F}_2({\tt{t}}) = cosec(t-0.5)cosec(2t)+cosec(3t)$

$\mathcal{F}_3({\tt{t}}) = cosec(t-1)cosec(t-0.5)cosec(2t)+cosec(t-1)cosec(3t)+cosec(t-0.5)cosec(2t)$, etc. Further-more, using (2.4), we have the following proportional $\alpha$-derivative of the co-secant Fibonacci polynomials;

$\mathcal{F}_0^{\alpha}({\tt{t}}) = 0, \ \ \mathcal{F}_1^{\alpha}({\tt{t}}) = \alpha[-2cosec(2t)cot(2t)]+(1-\alpha)cosec(2t),$

$\mathcal{F}_2^{\alpha}({\tt{t}}) = \alpha[-2cosec(t-0.5)cot(t-0.5)cosec(2t)-2cosec(t-0.5)cosec(2t)cot(2t)-3cosec(3t)cot(3t)]+(1-\alpha)[cosec(t-0.5)cosec(2t)+cosec(3t)],$ and etc.

4.   Bifurcation behavior of $\overline{\theta({\tt{t}})}$-Fibonacci generating function

The objective of this section is to demonstrate the efficacy of the bifurcation analysis of the $\overline{\theta({\tt{t}})}$-Fibonacci generating function and primary findings presented in this paper by employing analysis from the following precise examples.

A discrete one-dimensional dynamical system is a system subjected to a single equation of this type

where $x\in \mathbb{Z}$ and $f$ is a function of $x$. The variable ${\tt{t}}$ is in general considered as the time, but in discrete systems the time takes only discrete values, so that it is possible to take ${\tt{t}}\in \mathbb{Z}$.

A generalized discrete two-dimensional dynamical system is a system subjected to a single equation of this type

where ${\tt{t}}\in \mathbb{R}$. When we reformulate Eq (4.2), we obtain a two-dimensional discrete system

A trajectory is a set $\{x({\tt{t}})\}_{\tt{t} = 0}^{\infty}$ of points satisfying the above equation (4.1). It is evident that the initial point $x_0 = x(0)$ determines the entire trajectory. The behaviour of the dynamical system is therefore given by all the trajectories $\{x({\tt{t}}) : x(0) = x_0\}$ for all initial values $x_0\in I$. When the value of the parameter changes continuously, the behaviour of the system may change in a discontinuous way. One says that a bifurcation occurs for an isolate value of the parameter at which the type of dynamic changes. In bifurcation analysis, the region of stable operation is determined through the search of Hopf bifurcation points. This gives an insight into how the variations in the system parameters influence region of stable operation. This knowledge can be effectively used by the system designers to ensure the stability of the actual system.

A bifurcation diagram is a traditional and visual way to look into how dynamical systems, difference equations, and differential equations change over time ^[10]. This tool is excellent for looking at how the system reacts to changes in parameters ^[19]. This diagram illustrates the system's different dynamic patterns and phase transitions by plotting the link between the system reaction and parameters ^[5,8]. This section employs bifurcation theory to determine the existence of the period-doubling (flip) bifurcation. We discuss the $\overline{\theta({\tt{t}})}$-Fibonacci generating function and investigate the bifurcation analysis of the $\overline{\theta({\tt{t}})}$-Fibonacci sequences.

$\overline{\theta({\tt{t}})}$-Fibonacci sequences are generated by

where $\theta_1({\tt{t}})$ and $\theta_2({\tt{t}})$ are any trigonometric and hyperbolic functions. If $\theta_1({\tt{t}}) = sin(b_1{\tt{t}})$ and $\theta_2({\tt{t}}) = sin(b_2{\tt{t}})$ in (4.3), then we have the Fibonacci generating function

$(ii).$ If $\theta_1({\tt{t}}) = cos(b_1{\tt{t}})$ and $\theta_2({\tt{t}}) = cos(b_2{\tt{t}})$ in (4.3), then we have the Fibonacci generating function

4.1.   Chaotic behavior of the fibonacci generating function

In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system. A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos.

Now, we consider the recurrent form of (4.3)

and we consider the recurrent form of (4.4)

We consider the recurrent form of (4.5)

We examine the orbit $\{t_r\}_{r = 0}^{\infty}$ for any point $t_0$ within the domain of the map.

Figure $4(a)$ displays a periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $sine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 2$ and $b_1\in[-8, 11].$ Figure $4(b)$ displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $cosine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 2$ and $b_1\in[-5, 5].$ Figure $4(c)$ displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $tangent$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 1$ and $b_1\in[-5, 1].$ Figure $4(d)$ displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $cosine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 1$ and $b_1\in[-10, 5].$

Figure $5(a)$ displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $sine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_ = 2$ and $b_1\in[0, 0.3].$ Figure $5(b)$ displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $cosine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 2$ and $b_1\in[0.58, 2].$ Figure $5(c)$ displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $tangent$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 1$ and $b_1\in[-2.2, -1.8].$ Figure $5(d)$ displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the $cosine$ function with the initial condition $t_0 = 0.8$ at the intrinsic growth bifurcation parameter $b_2 = 1$ and $b_1\in[-3, -2.1].$

5.   Conclusions

This paper deduced an inverse formula for the $\overline{\theta({\tt{t}})}$-Fibonacci sequence. The inverse of a generic difference (nabla) operator with trigonometric coefficients of order 2 was used to derive this formula. The results we have obtained regarding the $E^*$ and $N^*$ solutions, Fibonacci polynomials, and the proportional derivative of the generic difference equation with trigonometric coefficients of order 2 will be applied to our future research. Additionally, we have conducted a bifurcation analysis of the $\overline{\theta({\tt{t}})}$-Fibonacci generating function.

Author's contributions

R.P: Conceptualization; R.P, S.T.M.T, and I.K.: Data curation; R.P, S.T.M.T, I.K, and M.V.C.: Formal analysis; P.R, I.K, and M.V.C.: Investigation; R.P, S.T.M.T, and I.K.: Methodology; R.P and S.T.M.T.: Writing-original draft; R.P, I.K and M.V.C.: Writing-review and editing. All authors have read and agreed to the published version of the article.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

"La derivada fraccional generalizada, nuevos resultados y aplicaciones a desigualdades integrales" Cod UIO-077-2024. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).

Conflict of interest

The authors declare that they have no conflicts of interest.