Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

Chaeyoung Lee; Yunjae Nam; Minjoon Bang; Seokjun Ham; Junseok Kim; Chaeyoung Lee; Yunjae Nam; Minjoon Bang; Seokjun Ham; Junseok Kim

doi:10.3934/math.20241297

AIMS Mathematics

2024, Volume 9, Issue 10: 26671-26687. doi: 10.3934/math.20241297

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Research article Special Issues

Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

1.
Department of Mathematics, Kyonggi University, Suwon, 16227, Republic of Korea
2.
Program in Actuarial Science and Financial Engineering, Korea University, Seoul 02841, Republic of Korea
3.
Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

Received: 05 July 2024 Revised: 02 September 2024 Accepted: 11 September 2024 Published: 13 September 2024
MSC : 35R11, 80M20, 39A14

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.

Keywords:

normalized time-fractional diffusion equation,
finite difference method,
Gamma function

Citation: Chaeyoung Lee, Yunjae Nam, Minjoon Bang, Seokjun Ham, Junseok Kim. Numerical investigation of the dynamics for a normalized time-fractional diffusion equation[J]. AIMS Mathematics, 2024, 9(10): 26671-26687. doi: 10.3934/math.20241297

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Abstract

1. Introduction and preliminaries

Let $\mathcal{A}$ indicate an analytic functions family, which is normalized under the condition $f$ $(0) =$ $f^{\prime}(0)-1 = 0$ in $\mathbb{U} = \{z:z\in \mathbb{C}$ and $\left \vert z \ \right \vert < 1\}$ and given by the following Taylor-Maclaurin series:

$\begin{equation} f \ (z) = z+\sum \limits_{n = 2}^{\infty}a_{n}z^{n}\ .\ \ \ \ \ \end{equation}$

(1.1)

Further, by $\mathcal{S}$ we shall denote the class of all functions in $\mathcal{A}$ which are univalent in $\mathbb{U}$ .

With a view to recalling the principle of subordination between analytic functions, let the functions $f$ and $g$ be analytic in $\mathbb{U}$ . Then we say that the function $f$ is subordinate to $g$ if there exists a Schwarz function $w\left(z\right)$ , analytic in $\mathbb{U}$ with

$\begin{equation*} \omega \left( 0\right) = 0, \ \left\vert \omega (z)\right\vert < 1, \ \left( z\in \mathbb{U}\right) \end{equation*}$

such that

$\begin{equation*} f \ \left( z\right) = g \ (\omega \left( z\right) ). \end{equation*}$

We denote this subordination by

$\begin{equation*} f\prec g\text{ or}\ f \ (z)\prec g \ (z). \end{equation*}$

In particular, if the function $g$ is univalent in $\mathbb{U}$ , the above subordination is equivalent to

$\begin{equation*} f \ (0) = g \ (0), \ f \ (\mathbb{U})\subset g \ ( \mathbb{U}). \end{equation*}$

The Koebe-One Quarter Theorem ^[11] asserts that image of $\mathbb{U}$ under every univalent function $f\in \mathcal{A}$ contains a disc of radius $\frac{1}{4}.\$ thus every univalent function $f$ has an inverse $\ f^{-1}\$ satisfying $\ f^{-1}(f(z)) = z$ and $f$ $(\ f^{-1}$ $(w)) = w$ $(\left\vert w\right\vert < r$ $_{0}(f$ $), r$ $_{0}(f$ $)$ $> \frac{ 1}{4}$ $)$ , where

$\begin{equation} \ f^{-1}(w) = w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{3}-5a_{2}a_{3}+a_{4})w^{4}+\cdots . \end{equation}$

(1.2)

A function $f\in \mathcal{A}$ is said to be bi-univalent functions in $\mathbb{U}$ if both $f$ and $\ f^{-1}$ are univalent in $\mathbb{U}$ . A function $f\in \mathcal{S}$ is said to be bi-univalent in $\mathbb{U}$ if there exists a function $g\in \mathcal{S}$ such that $g(z)$ is an univalent extension of $f^{-1}$ to $\mathbb{U}$ . Let $\Lambda$ denote the class of bi-univalent functions in $\mathbb{U}$ . The functions $\frac{z}{ 1-z}$ , $-\log (1-z)$ , $\frac{1}{2}\log \left(\frac{1+z}{1-z}\right)$ are in the class $\Lambda$ (see details in ^[20]). However, the familiar Koebe function is not bi-univalent. Lewin ^[17] investigated the class of bi-univalent functions $\Lambda$ and obtained a bound $|a_{2}|\leq 1.51$ . Motivated by the work of Lewin ^[17], Brannan and Clunie ^[9] conjectured that $|a_{2}|\leq \sqrt{2}.$ The coefficient estimate problem for $|a_{n}|\, \, \, \, (n\in \mathbb{N}, \, \, \, \, n\geq 3)$ is still open (^[20]). Brannan and Taha ^[10] also worked on certain subclasses of the bi-univalent function class $\Lambda$ and obtained estimates for their initial coefficients. Various classes of bi-univalent functions were introduced and studied in recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al. ^[20]. Motivated by this, many researchers ^[1], ^[4,5,6,7,8], ^[13,14,15], ^[20], ^[21], and ^[27,28,29], also the references cited there in) recently investigated several interesting subclasses of the class $\Lambda$ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. Recently, many researchers have been exploring bi-univalent functions, few to mention Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, Pell polynomials, Lucas–Lehmer polynomials, orthogonal polynomials and the other special polynomials and their generalizations are of great importance in a variety of branches such as physics, engineering, architecture, nature, art, number theory, combinatorics and numerical analysis. These polynomials have been studied in several papers from a theoretical point of view (see, for example, ^{[23,24,25,26,27,28,29,30]} also see references therein).

We recall the following results relevant for our study as stated in ^[3].

Let $p(x)$ and $q(x)$ be polynomials with real coefficients. The $(p, q)-$ Lucas polynomials $\mathcal{L}_{p, q, n}(x)$ are defined by the recurrence relation

$\begin{equation*} \mathcal{L}_{p, q, n}(x) = p(x)\mathcal{L}_{p, q, n-1}(x)+q(x)\mathcal{L} _{p, q, n-2}(x)\quad (n\geq 2), \end{equation*}$

from which the first few Lucas polynomials can be found as

$\begin{eqnarray} \mathcal{L}_{p, q, 0}(x) & = &2, \\ \mathcal{L}_{p, q, 1}(x) & = &p(x), \\ \mathcal{L}_{p, q, 2}(x) & = &p^{2}(x)+2q(x), \\ \mathcal{L}_{p, q, 3}(x) & = &p^{3}(x)+3p(x)q(x), ...\text{.} \end{eqnarray}$

(1.3)

For the special cases of $p(x)$ and $q(x),$ we can get the polynomials given $\mathcal{L}_{x, 1, n}(x)\equiv \mathcal{L}_{n}(x)$ Lucas polynomials, $\mathcal{L}_{2x, 1, n}(x)\equiv \mathcal{D}_{n}(x)$ Pell–Lucas polynomials, $\mathcal{L}_{1, 2x, n}(x)\equiv j_{n}(x)$ Jacobsthal–Lucas polynomials, $\mathcal{L}_{3x, -2, n}(x)\equiv F_{n}(x)$ Fermat–Lucas polynomials, $\mathcal{L}_{2x, -1, n}(x)\equiv T_{n}(x)$ Chebyshev polynomials first kind.

Lemma 1.1. ^[16] Let $G\{ \mathcal{L}(x)\}(z)$ be the generating function of the $(p, q)-$ Lucas polynomial sequence $\mathcal{L}_{p, q, n}(x).$ Then,

$\begin{equation*} G\{ \mathcal{L}(x)\}(z) = \sum\limits_{n = 0}^{\infty }\mathcal{L}_{p, q, n}(x)z^{n} = \frac{2-p(x)z}{1-p(x)z-q(x)z^{2}} \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{\{ \mathcal{L}(x)\}}(z) = G\{ \mathcal{L}(x)\}(z)-1 = 1+ \sum\limits_{n = 1}^{\infty }\mathcal{L}_{p, q, n}(x)z^{n} = \frac{1+q(x)z^{2}}{ 1-p(x)z-q(x)z^{2}}. \end{equation*}$

Definition 1.2. ^[22] For $\vartheta \geq 0,$ $\delta \in \mathbb{R},$ $\vartheta +i\delta \neq 0$ and $f\in \mathcal{A}$ , let $\mathcal{B} (\vartheta, \delta)$ denote the class of Bazilevič function if and only if

$\begin{equation*} Re\left[ {\left( {\frac{{z{f^\prime }(z)}}{{f(z)}}} \right){{\left( {\frac{{f(z)}}{z}} \right)}^{\vartheta + i\delta }}} \right] > 0\;. \end{equation*}$

Several authors have researched different subfamilies of the well-known Bazilevič functions of type $\vartheta$ from various viewpoints (see ^[3] and ^[19]). For Bazilevič functions of order $\vartheta +i\delta$ , there is no much work associated with Lucas polynomials in the literature. Initiating an exploration of properties of Lucas polynomials associated with Bazilevič functions of order $\vartheta +i\delta$ is the main goal of this paper. To do so, we take into account the following definitions. In this paper motivated by the very recent work of Altinkaya and Yalcin ^[3] (also see ^[18]) we define a new class $\mathcal{B}(\vartheta, \delta)$ , bi-Bazilevič function of $\Lambda$ based on $(p, q)-$ Lucas polynomials as below:

Definition 1.3. For $f\in \Lambda$ , $\vartheta \geq0,$ $\delta \in \mathbb{R},$ $\vartheta+i\delta \neq0$ and let $\mathcal{B}(\vartheta, \delta)$ denote the class of Bi-Bazilevič functions of order $t\ \$ and type $\vartheta+i\delta$ if only if

$\begin{equation} \left[ \left( \frac{zf^{\prime}(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta+i\delta}\right] \prec \mathcal{G}_{\{ \mathcal{L} (x)\}}(z)\quad (z\in \mathbb{U}) \end{equation}$

(1.4)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w}% \right) ^{\vartheta+i\delta}\right] \prec \mathcal{G}_{\{ \mathcal{L}% (x)\}}(w)\quad (w\in \mathbb{U}), \label{ieq4} \end{equation}$

(1.5)

where $\mathcal{G}_{\mathcal{L}_{p, q, n}} (z) \in \Phi$ and the function $g$ is described as $g(w) = f^{-1}(w).$

Remark 1.4. We note that for $\delta = 0$ the class $R(\vartheta, 0) = R(\vartheta)$ is defined by Altinkaya and Yalcin ^[2].

The class $\mathcal{B}(0, 0) = \mathcal{S}_\Lambda^*$ is defined as follows:

Definition 1.5. A function $f\in \Lambda$ is said to be in the class $\mathcal{S }_\Lambda^*$ , if the following subordinations hold

$\begin{equation*} \frac{zf^{\prime}(z)}{f(z)}\prec \mathcal{G}_{\{ \mathcal{L}(x)\}}(z)(z\in \mathbb{U}) \end{equation*}$

and

$\begin{equation*} \frac{wg^{\prime }(w)}{g(w)}\prec \mathcal{G}_{\{ \mathcal{L}(x)\}}(w)(w\in \mathbb{U}) \end{equation*}$

where $g(w) = f^{-1}(w).$

We begin this section by finding the estimates of the coefficients $|a_{2}|$ and $|a_{3}|$ for functions in the class $\mathcal{B}(\vartheta, \delta)$ .

2. Coefficient bounds for the function class $\mathcal{B}(\vartheta, \delta)$

Theorem 2.1. Let the function $f(z)$ given by 1.1 be in the class $\mathcal{B}(\vartheta, \delta)$ . Then

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{ \sqrt{\vert \{ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)-2\left( \vartheta +i\delta +1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +i\delta +1\right) ^{2} \vert }}. \end{equation*}$

and

$\begin{equation*} \left \vert a_{3}\right \vert \leq \frac{p^{2}(x)}{(\vartheta +1)^2+ \delta^2 } +\frac{p(x)}{\sqrt {(\vartheta +2)^2+ \delta^2} } . \end{equation*}$

Proof. Let $f\in \mathcal{B}(\vartheta, \delta, x)$ there exist two analytic functions $u, v:\mathbb{U}\rightarrow \mathbb{U}$ with $u(0) = 0 = v(0)$ , such that $|u(z)| < 1$ , $|v(w)| < 1,$ we can write from (1.4) and (1.5), we have

$\begin{equation} \left[ \left( \frac{zf^{\prime }(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta +i\delta }\right] = \mathcal{G}_{\{ \mathcal{L} (x)\}}(z)\quad (z\in \mathbb{U}) \quad \end{equation}$

(2.1)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w} \right) ^{\vartheta+i\delta}\right] = \mathcal{G}_{\{ \mathcal{L} (x)\}}(w)\quad (w\in \mathbb{U}), \end{equation}$

(2.2)

It is fairly well known that if

$\begin{equation*} |u(z)| = |u_{1}z+u_{2}z^{2}+\cdots | < 1 \end{equation*}$

and

$\begin{equation*} |v(w)| = |v_{1}w+v_{2}w^{2}+\cdots | < 1. \end{equation*}$

then

$\begin{equation*} |u_{k}|\leq 1\qquad and\qquad |v_{k}|\leq 1\quad (k\in \mathbb{N}) \end{equation*}$

It follows that, so we have

$\begin{eqnarray} \mathcal{G}_{\{ \mathcal{L}(x)\}}(u(z)) & = &1+\mathcal{L}_{p, q, 1}(x)u(z)+ \mathcal{L}_{p, q, 2}(x)u^{2}(z)+\ldots \\ & = &1+\mathcal{L}_{p, q, 1}(x)u_{1}z+[\mathcal{L}_{p, q, 1}(x)u_{2}+\mathcal{L} _{p, q, 2}(x)u_{1}^{2}]z^{2}+\ldots \end{eqnarray}$

(2.3)

and

$\begin{eqnarray} \mathcal{G}_{\{ \mathcal{L}(x)\}}(v(w)) & = &1+\mathcal{L}_{p, q, 1}(x)v(w)+ \mathcal{L}_{p, q, 2}(x)v^{2}(w)+\ldots \\ & = &1+\mathcal{L}_{p, q, 1}(x)v_{1}w+[\mathcal{L}_{p, q, 1}(x)v_{2}+\mathcal{L} _{p, q, 2}(x)v_{1}^{2}]w^{2}+\ldots \end{eqnarray}$

(2.4)

From the equalities (2.1) and (2.2), we obtain that

$\begin{equation} \left[ \left( \frac{zf^{\prime }(z)}{f(z)}\right) \left( \frac{f(z)}{z} \right) ^{\vartheta +i\delta }\right] = 1+\mathcal{L}_{p, q, 1}(x)u_{1}z+[ \mathcal{L}_{p, q, 1}(x)u_{2}+\mathcal{L}_{p, q, 2}(x)u_{1}^{2}]z^{2}+\ldots, \end{equation}$

(2.5)

and

$\begin{equation} \left[ \left( \frac{zg^{\prime}(w)}{g(w)}\right) \left( \frac{g(w)}{w} \right) ^{\vartheta+i\delta}\right] = 1+\mathcal{L}_{p, q, 1}(x)v_{1}w+[ \mathcal{L}_{p, q, 1}(x)v_{2}+\mathcal{L}_{p, q, 2}(x)v_{1}^{2}]w^{2}+\ldots, \end{equation}$

(2.6)

It follows from (2.5) and (2.6) that

$\begin{equation} \left( \vartheta +i\delta +1\right) a_{2} = \mathcal{L}_{p, q, 1}(x)u_1, , \end{equation}$

(2.7)

$\begin{equation} \frac{\left( \vartheta +i\delta -1\right) \left( \vartheta +i\delta +2\right) }{2}a_{2}^{2}-\left( \vartheta +i\delta +2\right) a_3 = \mathcal{L} _{p, q, 1}(x)u_2+\mathcal{L}_{p, q, 2}(x)u_1^2, \end{equation}$

(2.8)

and

$\begin{equation} -\left( \vartheta+i\delta+1\right) a_{2} = \mathcal{L}_{p, q, 1}(x)v_1, \end{equation}$

(2.9)

$\begin{equation} \frac{\left( \vartheta+i\delta+2\right) \left( \vartheta+i\delta+3\right) }{2 }a_{2}^{2}+\left( \vartheta+i\delta+2\right) a_3 = \mathcal{L}_{p, q, 1}(x)v_2+ \mathcal{L}_{p, q, 2}(x)v_1^2, \end{equation}$

(2.10)

From (2.7) and (2.9)

$\begin{equation} u_{1} = -v_{1} \end{equation}$

(2.11)

and

$\begin{equation} 2\left( \vartheta+i\delta+1\right) ^{2}a_{2}^{2} = \mathcal{L} _{p, q, 1}^{2}(x)(u_{1}^{2}+v_{1}^{2})., \end{equation}$

(2.12)

by adding (2.8) to (2.10), we get

$\begin{equation} \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right) a_{2}^{2} = \mathcal{L}_{p, q, 1}(x)(u_{2}+v_{2})+\mathcal{L} _{p, q, 2}(x)(u_{1}^{2}+v_{1}^{2}), \end{equation}$

(2.13)

by using (2.12) in equality (2.13), we have

$\begin{eqnarray} \left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right) -\frac{2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}}{\mathcal{L}^{2}_{p, q, 1}(x)}\right] a_{2}^{2}& = & \mathcal{L}_{p, q, 1}(x)(u_{2}+v_{2}), \end{eqnarray}$

$\begin{eqnarray} a_{2}^{2}& = &\frac{\mathcal{L}^3_{p, q, 1}(x)(u_{2}+v_{2})} {\left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L}^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}\right]}. \end{eqnarray}$

(2.14)

Thus, from (1.3) and (2.14) we get

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{\sqrt{\vert \{ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)-2\left( \vartheta +i\delta +1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +i\delta +1\right) ^{2} \vert }}. \end{equation*}$

Next, in order to find the bound on $|a_{3}|$ , by subtracting (2.10) from (2.8), we obtain

$\begin{eqnarray} 2\left( \vartheta +i\delta +2\right) a_{3}-2\left( \vartheta +i\delta +2\right) a_{2}^{2}& = &\mathcal{L}_{p, q, 1}(x)(u_{2}-v_{2})+\mathcal{L} _{p, q, 2}(x)(u_{1}^{2}-v_{1}^{2}) \\ 2\left( \vartheta +i\delta +2\right) a_{3}& = &\mathcal{L} _{p, q, 1}(x)(u_{2}-v_{2})+2\left( \vartheta +i\delta +2\right) a_{2}^{2} \\ a_{3}& = &\frac{\mathcal{L}_{p, q, 1}(x)(u_{2}-v_{2})}{2\left( \vartheta +i\delta +2\right)}+a_{2}^{2} \end{eqnarray}$

(2.15)

Then, in view of (2.11) and (2.12), we have from (2.15)

$\begin{equation*} a_{3} = \frac{\mathcal{L}_{p, q, 1}^{2}(x)}{2\left( \vartheta +i\delta +2\right) ^{2}}(u_{1}^{2}+v_{1}^{2})+\frac{\mathcal{L}_{p, q, 1}(x)}{2\left( \vartheta +i\delta +2\right) }(u_{2}-v_{2}). \end{equation*}$

$\begin{eqnarray*} |a_{3}|&\leq&\frac{p^{2}(x)}{\left|\vartheta +i\delta +1\right| ^{2}}+\frac{ p(x)}{|\vartheta +i\delta +2| } \notag \\ & = &\frac{p^{2}(x)}{(\vartheta +1)^2+ \delta^2 } +\frac{p(x)}{\sqrt { (\vartheta +2)^2+ \delta^2} } \end{eqnarray*}$

This completes the proof.

Taking $\delta = 0,$ in Theorem 2.1, we get the following corollary.

Corollary 2.2. Let the function $f(z)$ given by (1.1) be in the class $\mathcal{B} (\vartheta)$ . Then

$\begin{equation*} \left \vert a_{2}\right \vert \leq \frac{p(x)\sqrt {2p(x)}}{\sqrt{\vert \{ \left( \vartheta ^{2}+3\vartheta +2\right)-2\left( \vartheta+1\right) ^{2}\}p^2(x)-4q(x)\left( \vartheta +1\right) ^{2} \vert }} \end{equation*}$

and

$\begin{equation*} \left \vert a_{3}\right \vert \leq \frac{p^{2}(x)}{(\vartheta +2)^2} +\frac{ p(x)}{\vartheta +2} \end{equation*}$

Also, taking $\vartheta = 0$ and $\delta = 0,$ in Theorem 2.1, we get the results given in ^[18].

3. Fekete-Szegö inequality for the class $\mathcal{B}(\vartheta, \delta)$

Fekete-Szegö inequality is one of the famous problems related to coefficients of univalent analytic functions. It was first given by ^[12], the classical Fekete-Szegö inequality for the coefficients of $f\in \mathcal{S}$ is

$\begin{equation*} \left \vert a_{3}-\mu a_{2}^{2}\right \vert \leq1+2\exp \left( -2\mu /(1-\mu)\right) \text{ for }\mu \in \left[ 0, 1\right) . \end{equation*}$

As ${\mu} \rightarrow1^{-}$ , we have the elementary inequality $\left \vert a_{3}-a_{2}^{2}\right \vert \leq1$ . Moreover, the coefficient functional

$\begin{equation*} \varsigma_{\mu}(f) = a_{3}-\mu a_{2}^{2} \end{equation*}$

on the normalized analytic functions $f$ in the unit disk $\mathbb{U}$ plays an important role in function theory. The problem of maximizing the absolute value of the functional $\varsigma {\mu} (f)$ is called the Fekete-Szegö problem.

In this section, we are ready to find the sharp bounds of Fekete-Szegö functional $\varsigma _{\mu }(f)$ defined for $f\in \mathcal{B}(\vartheta, \delta)$ given by (1.1).

Theorem 3.1. Let f given by (1.1) be in the class $\mathcal{B} (\vartheta, \delta)$ and ${\mu} \in \mathbb{R}$ . Then

$\begin{equation*} \left \vert a_{3}- {\mu} a_{2}^{2}\right \vert \leq \left \{ \begin{array}{c} \frac{p(x)}{\sqrt{(\vartheta+2)^{2}+\delta ^{2}}}, \ \ \ \ \ \ \ \ 0\leq \left \vert h(\mu )\right \vert \leq \frac{1}{2\sqrt{ (\vartheta+2)^{2}+\delta ^{2}}} \\ 2p(x)\left \vert h(\mu )\right \vert , \ \ \ \ \ \ \ \ \ \ \ \ \ \left \vert h(\mu )\right \vert \geq \frac{1}{2\sqrt{(\vartheta+2)^{2}+ \delta ^{2}}} \end{array} \right. \end{equation*}$

where

$\begin{equation*} h(\mu) = \frac{\mathcal{L}^2_{p, q, 1}(x)(1-\mu)}{\left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L} ^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}}. \end{equation*}$

Proof. From (2.14) and (2.15), we conclude that

$\begin{eqnarray*} a_{3}- {\mu} a_{2}^{2}& = &(1-\mu )\frac{\mathcal{L}^3_{p, q, 1}(x)(u_{2}+v_{2})} {\left[ \left( \left( \vartheta +i\delta \right) ^{2}+3\left( \vartheta +i\delta \right) +2\right)\mathcal{L}^{2}_{p, q, 1}(x)-2\mathcal{L}_{p, q, 2}(x)\left( \vartheta +i\delta +1\right) ^{2}\right]} \\ &+&\frac{\mathcal{L}_{p, q, 1}(x)}{2\left( \vartheta +i\delta +2\right) } (u_{2}-v_{2}) \end{eqnarray*}$

$\begin{eqnarray*} = \mathcal{L}_{p, q, 1}(x)\left[ \left( h(\mu )+\frac{1}{2(\vartheta +i\delta +2)}\right) u_{2}+\left( h(\mu )-\frac{1}{2(\vartheta +i\delta +2)}\right) v_{2}\right] \end{eqnarray*}$

where

Then, in view of (1.3), we obtain

We end this section with some corollaries.

Taking ${\mu} = 1$ in Theorem 3.1, we get the following corollary.

Corollary 3.2. If $f\in \mathcal{B}(\vartheta, \delta)$ , then

$\begin{equation*} \left \vert a_{3}-a_{2}^{2}\right \vert \leq \frac{p(x)}{\sqrt{ (\vartheta+2)^{2}+\delta ^{2}}}. \end{equation*}$

Taking $\delta = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.3. Let $f$ given by (1.1) be in the class $\mathcal{B}(\vartheta, 0)$ . Then

$\begin{equation*} \left \vert a_{3}- {\mu} a_{2}^{2}\right \vert \leq \left \{ \begin{array}{c} \frac{p(x)}{\vartheta+2}, \ \ \ \ \ \ \ \ 0\leq \left \vert h(\mu )\right \vert \leq \frac{1}{2(\vartheta+2)} \\ 2p(x)\left \vert h(\mu )\right \vert , \ \ \ \ \ \ \ \ \ \ \ \ \ \left \vert h(\mu )\right \vert \geq \frac{1}{2(\vartheta+2)} \end{array} \right. \end{equation*}$

Also, taking $\vartheta = 0,$ $\delta = 0$ and ${\mu} = 1$ in Theorem 3.1, we get the following corollary.

Corollary 3.4. Let $f$ given by (1.1) be in the class $\mathcal{B}$ . Then

$\begin{equation*} \left| {{a_3} - a_2^2} \right| \le \frac{{p(x)}}{2}. \end{equation*}$

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	J. J. Liu, M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 89 (2010), 1769–1788. https://doi.org/10.1080/00036810903479731 doi: 10.1080/00036810903479731
[2]	L. Feng, I. Turner, P. Perré, K. Burrage, The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects, J. Comput. Phys., 480 (2023), 112020. https://doi.org/10.1016/j.jcp.2023.112020 doi: 10.1016/j.jcp.2023.112020
[3]	M. Biglari, A. R. Soheili, Efficient simulation of two-dimensional time-fractional Navier–Stokes equations using RBF-FD approach, Eng. Anal. Bound. Elem., 160 (2024), 134–159. https://doi.org/10.1016/j.enganabound.2023.12.021 doi: 10.1016/j.enganabound.2023.12.021
[4]	F. A. Rihan, Q. M. Al-Mdallal, H. J. AlSakaji, A. Hashish, A fractional-order epidemic model with time-delay and nonlinear incidence rate, Chaos Soliton. Fract., 126 (2019), 97–105. https://doi.org/10.1016/j.chaos.2019.05.039 doi: 10.1016/j.chaos.2019.05.039
[5]	M. Inc, The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476–484. https://doi.org/10.1016/j.jmaa.2008.04.007 doi: 10.1016/j.jmaa.2008.04.007
[6]	J. G. Liu, J. Zhang, A new approximate method to the time fractional damped Burger equation, AIMS Math., 8 (2023), 13317–13324. https://doi.org/10.3934/math.2023674 doi: 10.3934/math.2023674
[7]	A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, W. Weera, Evaluation of time-fractional Fisher's equations with the help of analytical methods, AIMS Math., 7 (2022), 18746–66. https://doi.org/10.3934/math.20221031 doi: 10.3934/math.20221031
[8]	X. Qin, X. Yang, P. Lyu, A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation, AIMS Math., 6 (2021), 11449–11466. https://doi.org/10.3934/math.2021663 doi: 10.3934/math.2021663
[9]	W. Chen, X. Xu, S. P. Zhu, Analytically pricing double barrier options based on a time-fractional Black–Scholes equation, Comput. Math. Appl., 69 (2015), 1407–1419. https://doi.org/10.1016/j.camwa.2015.03.025 doi: 10.1016/j.camwa.2015.03.025
[10]	A. Golbabai, O. Nikan, T. Nikazad, Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market, Comput. Appl. Math., 38 (2019), 1–24. https://doi.org/10.1007/s40314-019-0957-7 doi: 10.1007/s40314-019-0957-7
[11]	H. Zhang, F. Liu, I. Turner, Q. Yang, Numerical solution of the time fractional Black–Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772–1783. https://doi.org/10.1016/j.camwa.2016.02.007 doi: 10.1016/j.camwa.2016.02.007
[12]	Q. Du, J. Yang, Z. Zhou, Time-fractional Allen–Cahn equations: analysis and numerical methods, J. Sci. Comput., 85 (2020), 42. https://doi.org/10.1007/s10915-020-01351-5 doi: 10.1007/s10915-020-01351-5
[13]	H. Liu, A. Cheng, H. Wang, J. Zhao, Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation, Comput. Math. Appl., 76 (2018), 1876–1892. https://doi.org/10.1016/j.jocs.2023.102114 doi: 10.1016/j.jocs.2023.102114
[14]	B. Derbissaly, M. Sadybekov, Inverse source problem for multi-term time-fractional diffusion equation with nonlocal boundary conditions, AIMS Math., 9 (2024), 9969–9988. https://doi.org/10.3934/math.2024488 doi: 10.3934/math.2024488
[15]	W. M. Abd-Elhameed, H. M. Ahmed, Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials, AIMS Math., 9 (2024), 2137–2166. https://doi.org/10.3934/math.2024107 doi: 10.3934/math.2024107
[16]	Y. E. Aghdam, H. Mesgarani, Z. Asadi, V. T. Nguyen, Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model, AIMS Math., 8 (2023), 29474. https://doi.org/10.3934/math.20231509 doi: 10.3934/math.20231509
[17]	J. Kim, S. Kwak, H. G. Lee, Y. Hwang, S. Ham, A maximum principle of the Fourier spectral method for diffusion equations, Electron. Res. Arch., 31 (2023), 5396–5405. https://doi.org/10.3934/era.2023273 doi: 10.3934/era.2023273
[18]	J. M. Carcione, Theory and modeling of constant-Q P-and S-waves using fractional time derivatives, Geophysics, 74 (2009), T1–T11. https://doi.org/10.1190/1.3008548 doi: 10.1190/1.3008548
[19]	J. M. Carcione, F. Cavallini, F. Mainardi, A. Hanyga, Time-domain modeling of constant-Q seismic waves using fractional derivatives, Pure Appl. Geophys., 159 (2002), 1719–1736. https://doi.org/10.1007/s00024-002-8705-z doi: 10.1007/s00024-002-8705-z
[20]	S. Ham, J. Kim, Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation, Math. Comput. Simul., 207 (2023), 453–465. https://doi.org/10.1016/j.matcom.2023.01.016 doi: 10.1016/j.matcom.2023.01.016
[21]	J. Wang, Z. Han, W. Jiang, J. Kim, A fast, efficient, and explicit phase-field model for 3D mesh denoising, Appl. Math. Comput., 458 (2023), 128239. https://doi.org/10.1016/j.amc.2023.128239 doi: 10.1016/j.amc.2023.128239
[22]	J. W. Thomas, Numerical partial differential equations: finite difference methods in Springer Science & Business Media (2013).
[23]	M. Sarboland, A. Aminataei, On the numerical solution of time fractional Black-Scholes equation, Int. J. Comput. Math., 99 (2022), 1736–1753. https://doi.org/10.1080/00207160.2021.2011248 doi: 10.1080/00207160.2021.2011248
[24]	J. Huang, Z. Cen, J. Zhao, An adaptive moving mesh method for a time-fractional Black–-Scholes equation, Adv. Differ. Equ., 2019 (2019), 1–14. https://doi.org/10.1186/s13662-019-2453-1 doi: 10.1186/s13662-019-2453-1
[25]	B. Xia, R. Yu, X. Song, X. Zhang, J. Kim, An efficient data assimilation algorithm using the Allen–Cahn equation, Eng. Anal. Bound. Elem., 155 (2023), 511–517. https://doi.org/10.1016/j.enganabound.2023.06.029 doi: 10.1016/j.enganabound.2023.06.029
[26]	Y. Hwang, I. Kim, S. Kwak, S. Ham, S. Kim, J. Kim, Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation, Electron. Res. Arch., 31 (2023), 5104–5123. https://doi.org/10.3934/era.2023261 doi: 10.3934/era.2023261
[27]	Y. Hwang, S. Ham, C. Lee, G. Lee, S. Kang, J. Kim, A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes, Electron. Res. Arch., 31 (2023), 4557–4578. https://doi.org/10.3934/era.2023233 doi: 10.3934/era.2023233
[28]	C. Lee, S. Kim, S. Kwak, Y. Hwang, S. Ham, S. Kang, J. Kim, Semi-automatic fingerprint image restoration algorithm using a partial differential equation, AIMS Math., 8 (2023), 27528-27541. https://doi.org/10.3934/math.20231408 doi: 10.3934/math.20231408
[29]	Z. W. Fang, H. W. Sun, H. Wang, A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations, Comput. Math. Appl., 80 (2020), 1443–1458. https://doi.org/10.1016/j.camwa.2020.07.009 doi: 10.1016/j.camwa.2020.07.009

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