Transfer learning for robust urban network-wide traffic volume estimation with uncertain detector deployment scheme

Jiping Xing; Yunchi Wu; Di Huang; Xin Liu; Jiping Xing; Yunchi Wu; Di Huang; Xin Liu

doi:10.3934/era.2023011

Electronic Research Archive

2023, Volume 31, Issue 1: 207-228. doi: 10.3934/era.2023011

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Research article Special Issues

Transfer learning for robust urban network-wide traffic volume estimation with uncertain detector deployment scheme

1.
Jiangsu Key Laboratory of Urban ITS, Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, School of Transportation, Southeast University, China
2.
School of Public Administration, Huazhong University of Science and Technology, Wuhan, China

Received: 06 September 2022 Revised: 06 October 2022 Accepted: 13 October 2022 Published: 28 October 2022

Real-time and accurate network-wide traffic volume estimation/detection is an essential part of urban transport system planning and management. As it is impractical to install detectors on every road segment of the city network, methods on the network-wide flow estimation based on limited detector data are of considerable significance. However, when the plan of detector deployment is uncertain, existing methods are unsuitable to be directly used. In this study, a transfer component analysis (TCA)-based network-wide volume estimation model, considering the different traffic volume distributions of road segments and transforming traffic features into common data space, is proposed. Moreover, this study applied taxi GPS (global positioning system) data and cellular signaling data with the same spatio-temporal coverage to improve feature extraction. In numerical experiments, the robustness and stability of the proposed network-wide estimation method outperformed other baselines in the two subnetworks selected from the urban centers and suburbs.

Keywords:

Citation: Jiping Xing, Yunchi Wu, Di Huang, Xin Liu. Transfer learning for robust urban network-wide traffic volume estimation with uncertain detector deployment scheme[J]. Electronic Research Archive, 2023, 31(1): 207-228. doi: 10.3934/era.2023011

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Abstract

1. Introduction

Non-linear partial differential equations are extensively used in science and engineering to model real-world phenomena ^[1,2,3,4]. Using fractional operators like the Riemann-Liouville (RL) and the Caputo operators which have local and singular kernels, it is difficult to express many non-local dynamics systems. Thus to describe complex physical problems, fractional operators with non-local and non-singular kernels ^[5,6] were defined. The Atangana-Baleanu (AB) fractional derivative operator is one of these type of fractional operators which is introduced by Atangana and Baleanu^[7].

The time fractional Kolmogorov equations (TF-KEs) are defined as

$\begin{equation} {^{AB}\! D}_{t}^{\gamma} g(s, t) = \vartheta_{1}(s)D_{s} g(s, t)+\vartheta_{2}(s)D_{ss} g(s, t)+\omega \left(s, t\right), \quad 0 < \gamma \leq 1, \end{equation}$

(1.1)

with the initial and boundary conditions

$\begin{equation*} \begin{array}{ll} g(s, 0) = d_{0}(s), &\\ g(0, t) = d_{1}(t), & g(1, t) = d_{2}(t), \end{array} \end{equation*}$

where $(s, t)\in [0, 1]\times [0, 1]$ , ${^{AB}\! D}_{t}^{\gamma}$ denotes the Atangana-Baleanu (AB) derivative operator, $D_{s} g(s, t) = \dfrac{\partial}{\partial s}g(s, t)$ and $D_{ss} g(s, t) = \dfrac{\partial^{2}}{\partial s^{2}}g(s, t)$ . If $\vartheta_{1}(s)$ and $\vartheta_{2}(s)$ are constants, then Eq (1.1) is presenting the time fractional advection-diffusion equations (TF-ADEs).

Many researchers are developing methods to find the solution of partial differential equations of fractional order. Analytical solutions or formal solutions of such type of equations are difficult; therefore, numerical simulations of these equations inspire a large amount of attentions. High accuracy methods can illustrate the anomalous diffusion phenomenon more precisely. Some of the efficient techniques are Adomian decomposition ^[8,9], a two-grid temporal second-order scheme ^[10], the Galerkin finite element method ^[11], finite difference ^[12], a differential transform ^[13], the orthogonal spline collocation method ^[14], the optimal homotopy asymptotic method ^[15], an operational matrix (OM) ^{[16,17,18,19,20,21,22,23,24]}, etc.

The OM is one of the numerical tools to find the solution of a variety of differential equations. OMs of fractional derivatives and integration were derived using polynomials like the Chebyshev ^[16], Legendre ^[17,18], Bernstein ^[19], clique ^[20], Genocchi ^[21], Bernoulli ^[22], etc. In this work, with the help of the Hosoya polynomial (HS) of simple paths and OMs, we reduce problem (1.1) to the solution of a system of nonlinear algebraic equations, which greatly simplifies the problem under study.

The sections are arranged as follows. In Section 2, we review some basic preliminaries in fractional calculus and interesting properties of the HP. Section 3 presents a new technique to solve the TF-KEs. The efficiency and simplicity of the proposed method using examples are discussed in Section 5. In Section 6, the conclusion is given.

2. Preliminaries

In this section we discuss some basic preliminaries of fractional calculus and the main properties of the HP. We also compute an error bound for the numerical solution.

2.1. Fractional calculus

Definition 2.1. (See ^[25]) Let $0 < \gamma\leq 1$ . The RL integral of order $\gamma$ is defined as

$\begin{eqnarray*} {^{RL}\! I}_{s}^{\gamma} g(s) = \frac{1}{\Gamma(\gamma)}\int_{0}^{s} (s-\xi)^{\gamma-1} g(\xi)\ d\xi. \end{eqnarray*}$

One of the properties of the fractional order of RL integral is

$\begin{align*} &{^{R\!L}\! I}_{s}^{\gamma} s^{\upsilon} = \dfrac{\Gamma(\upsilon+1)}{\Gamma(\upsilon+1+\gamma)}s^{\upsilon+\gamma}, \quad \upsilon\geq 0. \end{align*}$

Definition 2.2. (See ^[7]) Let $0 < \gamma \leq 1$ , $g\in H^{1}(0, 1)$ and $\Phi(\gamma)$ be a normalization function such that $\Phi(0) = \Phi(1) = 1$ and $\Phi(\gamma) = 1-\gamma+\dfrac{\gamma}{\Gamma(\gamma)}$ . Then, the following holds

1) The AB derivative is defined as

$\begin{align*} &{^{AB}\! D}_{s}^{\gamma} g(s) = \frac{\Phi(\gamma)}{1-\gamma}\int_{0}^{s} E_{\gamma}(-\frac{\gamma}{1-\gamma}(s-\xi)^{\gamma}) g{'}(\xi)\ d\xi, \quad {0 < \gamma < 1}, \\ &{{^{AB}\! D}_{s}^{\gamma} g(s) = g'(s), \quad \gamma = 1}, \end{align*}$

where $E_{\gamma}(s) = \sum\limits_{j = 0}^{\infty}\frac{s^{j}}{\Gamma(\gamma j+1)}$ is the Mittag-Leffler function.

2) The AB integral is given as

$\begin{equation} {^{AB}\! I}_{s}^{\gamma} g(s) = \dfrac{1-\gamma}{\Phi(\gamma)}g(s)+\dfrac{\gamma}{\Phi(\gamma)\Gamma(\gamma)}\int_{0}^{s}(s-{\xi})^{\gamma-1}g(\xi)\, d\xi. \end{equation}$

(2.1)

Let $v_{\gamma} = \dfrac{1-\gamma}{\Phi(\gamma)}$ and $w_{\gamma} = \dfrac{1}{\Phi(\gamma)\Gamma(\gamma)}$ ; then, we can rewrite (2.1) as

$\begin{equation*} {^{AB}\! I}_{s}^{\gamma} g(s) = v_{\gamma}g(s)+w_{\gamma} \Gamma(\gamma+1){^{RL}\! I}_{s}^{\gamma}g(s). \end{equation*}$

The AB integral satisfies the following property ^[26]:

${^{AB}\! I_{s}^{\gamma}}({^{AB}\! D}_{s}^{\gamma} g(s)) = g(s)-g(0).$

2.2. The HP and their properties

In 1988, Haruo Hosoya introduced the concept of the HP ^[27,28]. This polynomial is used to calculate distance between vertices of a graph ^[29]. In ^[30,31], the HP of path graphs is obtained. The HP of the path graphs is described as

$\begin{align*} \tilde{ H}(G, s) = \sum\limits_{l\geq 0}d(G, l)s^{l}, \end{align*}$

where $d(G, l)$ denotes the distance between vertex pairs in the path graph ^[32,33]. Here we consider path graph with vertices $n$ where $n\in N$ . Based on $n$ vertex values the Hosoya polynomials are calculated ^[34]. Let us consider the path $P_{n}$ with $n$ vertices; then the HP of the $P_{i}, \, i = 1, 2, \cdots, n$ are computed as

$\begin{align} \begin{array}{lllll} \tilde{ H}(P_{1}, s)& = \sum\limits_{l\geq 0}d(P_{1}, l)s^{l} = 1, \nonumber\\ \tilde{ H}(P_{2}, s)& = \sum\limits_{l = 0}^1d(P_{2}, l)s^{l} = s+2, \nonumber\\ \tilde{ H}(P_{3}, s)& = \sum\limits_{l = 0}^2 d(P_{3}, l)s^{l} = s^{2}+2s+3, \nonumber\\ &\vdots\nonumber\\ \tilde{ H}(P_{n}, s)& = n+(n-1) s+(n-2)s^{2}+\cdots+\left(n-(n-2)\right)s^{n-2}+\left(n-(n-1)\right)s^{n-1}. \end{array} \end{align}$

Consider any function $g(s)$ in $L^{2}(0, 1)$ ; we can approximate it using the HP as follows:

$\begin{equation} g(s)\simeq \tilde{g}(s) = \sum\limits_{i = 1}^{N+1}h_{i}\ \tilde{ H}(P_{i}, s) = h^{T} \mathrm{H}(s), \end{equation}$

(2.2)

where

$\begin{equation*} h = [h_{1}, h_{2}, \cdots, h_{N+1}]^{T}, \end{equation*}$

and

$\begin{equation} \mathrm{H}(s) = [\tilde{ H}(P_{1}, s), \tilde{ H}(P_{2}, s), \cdots, \tilde{ H}(P_{N+1}, s)]^{T}. \end{equation}$

(2.3)

From (2.2), we have

$\begin{equation*} h = {\mathcal{Q}}^{-1}\langle g(s), H(s)\rangle, \end{equation*}$

where ${\mathcal{Q}} = \langle\mathrm{H}(s), \mathrm{H}(s)\rangle$ and $\langle \cdot, \cdot\rangle$ denotes the inner product of two arbitrary functions.

Now, consider the function $g(s, t) \in L^{2}\left([0, 1]\times [0, 1]\right)$ ; then, it can be expanded in terms of the HP by using the infinite series,

$\begin{equation} g(s, t) = \sum\limits_{i = 1}^{\infty}\sum\limits_{j = 1}^{\infty}h_{ij}\tilde{ H}(P_{i}, s)\tilde{ H}(P_{j}, t). \end{equation}$

(2.4)

If we consider the first $(N+1)^{2}$ terms in (2.4), an approximation of the function $g(s, t)$ is obtained as

$\begin{equation} g(s, t)\simeq\sum\limits_{i = 1}^{N+1}\sum\limits_{j = 1}^{N+1}h_{ij}\tilde{ H}(P_{i}, s)\tilde{ H}(P_{j}, t) = \mathrm{H}^{T}(s)\tilde{ h} \mathrm{H}(t), \end{equation}$

(2.5)

where

$\begin{equation*} \tilde{ h} = {\mathcal{Q}}^{-1}\, \langle \mathrm{H}(s), \langle g(s, t), \mathrm{H}(t)\rangle \rangle {\mathcal{Q}}^{-1}. \end{equation*}$

Theorem 2.1. The integral of the vector $\mathrm{H}(s)$ given by (2.3) can be approximated as

$\begin{eqnarray} \int_{0}^{s} \mathrm{H}(\xi)d\xi\simeq R \mathrm{H}(s), \end{eqnarray}$

(2.6)

where $R$ is called the OM of integration for the HP.

Proof. Firstly, we express the basis vector of the HP, $\mathrm{H}(s)$ , in terms of the Taylor basis functions,

$\begin{equation} \mathrm{H}(s) = \mathcal{A}\widehat{S}(s), \end{equation}$

(2.7)

where

$\begin{equation*} \widehat{S}(s) = \left[1, s, \cdots , s^{N}\right]^{T}, \end{equation*}$

and

$\mathcal{A} = \left[a_{q, r}\right], \quad q, r = 1, 2, \cdots, N+1,$

with

$\begin{equation*} a_{q, r} = \begin{cases} q-(r-1), & q\geq r, \\ 0, & q < r. \end{cases} \end{equation*}$

Now, we can write

$\begin{eqnarray*} \int_{0}^{s}\mathrm{H}(\xi)d\xi = \mathcal{A} \int_{0}^{s}\widehat{S}(\xi)d\xi = \mathcal{A} \mathcal{B} S^{*}(s), \end{eqnarray*}$

where $\mathcal{B} = [b_{q, r}], \; q, r = 1, 2, \cdots, N+1$ is an $(N+1)\times(N+1)$ matrix with the following elements

$\begin{eqnarray*} b_{q, r} = \left\{ \begin{array}{ll} \dfrac{1}{q}, &q = r, \\ 0, &q\neq r, \end{array} \right. \end{eqnarray*}$

and

$\begin{eqnarray*} S^{*}(s) = \begin{bmatrix} s, s^{2}, \cdots, s^{N+1} \end{bmatrix}^{T}. \end{eqnarray*}$

Now, by approximating $s^{k}, \; k = 1, 2, \cdots, N+1$ in terms of the HP and by (2.7), we have

$\begin{eqnarray*} \left\{ \begin{array}{ll} s^{k} = \mathcal{A}^{-1}_{k+1}\mathrm{H}(s), &k = 1, 2, \cdots, N, \\ s^{N+1} = \mathcal{L}^{T}\mathrm{H}(s), \end{array} \right. \end{eqnarray*}$

where $\mathcal{A}^{-1}_{r}$ , $r = 2, 3, \cdots, N+1$ is the $r$ -th row of the matrix $\mathcal{A}^{-1}$ and $\mathcal{L} = {\mathcal{Q}}^{-1}\langle s^{N+1}, \mathrm{H}(s)\rangle$ . Then, we get

$\begin{eqnarray*} S^{*}(s) = \mathcal{E} \mathrm{H}(s), \end{eqnarray*}$

where $\mathcal{E} = \begin{bmatrix}\mathcal{A}_{2}^{-1}, \mathcal{A}_{3}^{-1}, \cdots, \mathcal{A}_{N+1}^{-1}, \mathcal{L}^{T} \end{bmatrix}^{T}$ . Therefore, by taking $R = \mathcal{A}\mathcal{B}\mathcal{E}$ , the proof is completed.

Theorem 2.2. The OM of the product based on the HP is given by (2.3) can be approximated as

$\begin{align*} C^{T}\, \mathrm{H}(s)\, \mathrm{H}^{T}(s)\simeq \mathrm{H}^{T}(s)\, \widehat{C}, \end{align*}$

where $\widehat{C}$ is called the OM of product for the HP.

Proof. Multiplying the vector $C = [c_1, c_2, \cdots, c_{N+1}]^T$ by $\mathrm{H}(s)$ and $\mathrm{H}^{T}(s)$ gives

$\begin{align} C^{T}\mathrm{H}(s)\, \mathrm{H}^{T}(s)& = C ^{T}\mathrm{H}(s)\, (\widehat{S}^{T}(s)\, \mathcal{A}^{T})\\ & = \left[C ^{T}\mathrm{H}(s), s\, (C^{T}\mathrm{H}(s)), \cdots, s^{{N}}(C^{T}\mathrm{H}(s)) \right] \mathcal{A}^{T}\\ & = \left[ \sum\limits_{i = 1}^{N+1}c_{i}\, \tilde{ H}(P_{i}, s), \sum\limits_{i = 1}^{N+1}c_{i}\, s\, \tilde{ H}(P_{i}, s), \cdots, \sum\limits_{i = 1}^{N+1}c_{i}\, s^{N}\, \tilde{ H}(P_{i}, s)\right] \mathcal{A}^{T}. \end{align}$

(2.8)

Taking $e_{k, i} = \left[e_{k, i}^{1}, e_{k, i}^{2}, \cdots, e_{k, i}^{N+1} \right]^{T}$ and expanding $s^{k-1}\, \tilde{ H}(P_{i}, s)\simeq e_{k, i}^{T}\, \mathrm{H}(s), \; i, k = 1, 2, \cdots, N+1$ using the HP, we can write

$\begin{align*} e_{k, i}& = \mathcal{Q}^{-1}\int_{0}^{1}s^{k-1}\, \tilde{ H}(P_{i}, s)\mathrm{H}(s)\, ds\\ & = \mathcal{Q}^{-1}\left[ \int_{0}^{1}s^{k-1}\, \tilde{ H}(P_{i}, s)\, \tilde{ H}(P_{1}, s)\, ds, \int_{0}^{1}s^{k-1}\, \tilde{ H}(P_{i}, s)\, \tilde{ H}(P_{2}, s)\, ds, \cdots, \int_{0}^{1}s^{k-1}\, \tilde{ H}(P_{i}, s)\, \tilde{ H}(P_{N+1}, s)\, ds\right]^{T}. \end{align*}$

Therefore,

$\begin{align} \sum\limits_{i = 1}^{N+1}c_{i}\, s^{k-1}\, \tilde{H}(P_{i}, s)&\simeq \sum\limits_{i = 1}^{N+1}c_{i}\left( \sum\limits_{j = 1}^{N+1} e_{k, i}^{j}\, \tilde{H}(P_{j}, s)\right)\\ & = \sum\limits_{j = 1}^{N+1} \tilde{H}(P_{j}, {s})\left( \sum\limits_{i = 1}^{N+1} c_{i} \, e_{k, i}^{j}\right) \\ & = \mathrm{H}^{T}(s)\left[ \sum\limits_{i = 1}^{N+1}c_{i}\, e_{k, i}^{1}, \sum\limits_{i = 1}^{N+1}c_{i}\, e_{k, i}^{2}, \cdots, \sum\limits_{i = 1}^{N+1}c_{i}\, e_{k, i}^{N+1}\right]^{T} \\ & = \mathrm{H}^{T}(s)\left[e_{k, 1}, e_{k, 2}, \cdots, e_{k, N+1} \right]\, C\\ & = H^{T}(s)\, E_{k}\, C, \end{align}$

(2.9)

where $E_{k}$ is an $(N+1)\times(N+1)$ matrix and the vectors $e_{k, i}$ for $k = 1, 2, \cdots, N+1$ are the columns of $E_{k}$ . Let $\overline{E}_{k} = E_{k}\, C, \quad k = 1, 2, \cdots, N+1$ . Setting $\overline{C} = \left[\overline{E}_{1}, \overline{E}_{2}, \cdots, \overline{E}_{N+1} \right]$ as an $(N+1)\times(N+1)$ matrix and using (2.8) and (2.9), we have

$\begin{align*} C^{T} \mathrm{H}(s)\, \mathrm{H}^{T}(s)& = \left[ \sum\limits_{i = 1}^{N+1}c_{i}\, \tilde{H}(P_{i}, s), \sum\limits_{i = 1}^{N+1}c_{i}\, s\, \tilde{H}(P_{i}, s), \cdots, \sum\limits_{i = 1}^{N+1}c_{i}\, s^{N}\, \tilde{H}(P_{i}, s) \right]\, A^{T}\nonumber\\ &\simeq \mathrm{H}^{T}(s)\, \widehat{C}, \end{align*}$

where by taking $\widehat{C} = \overline{C}\, A^{T}$ , the proof is completed.

Theorem 2.3. Consider the given vector $\mathrm{H}(s)$ in (2.3); the fractional RL integral of this vector is approximated as

$\begin{equation*} {^{R\!L}\! I_{s}^{\gamma}}\mathrm{H}(s)\simeq \mathcal{P}^{\gamma} \mathrm{H}(s), \end{equation*}$

where $\mathcal{P}^{\gamma}$ is named the OM based on the HP which is given by

$\begin{equation*} \mathcal{P}^{\gamma} = \begin{bmatrix} \sigma_{1, 1, 1}&\sigma_{1, 2, 1}&\cdots&\sigma_{1, N+1, 1}\\ \sum\limits_{k = 1}^{2}\sigma_{2, 1, k}& \sum\limits_{k = 1}^{2}\sigma_{2, 2, k}&\cdots& \sum\limits_{k = 1}^{2}\sigma_{2, N+1, k}\\ \vdots&\vdots&\ddots&\vdots\\ \sum\limits_{k = 1}^{N+1}\sigma_{N+1, 1, k}& \sum\limits_{k = 1}^{N+1}\sigma_{N+1, 2, k}&\cdots& \sum\limits_{k = 1}^{N+1}\sigma_{N+1, N+1, k} \end{bmatrix}, \end{equation*}$

with

$\begin{equation*} \sigma_{i, j, k} = \dfrac{\left(i-(k-1)\right)\Gamma(k)e_{k, j}}{\Gamma(k+\gamma)}. \end{equation*}$

Proof. First, we rewrite $\tilde{H}(P_{i}, s)$ in the following form:

$\begin{align*} \tilde{H}(P_{i}, s) = \sum\limits_{k = 1}^{i}\left(i-(k-1)\right)s^{k-1}. \end{align*}$

Let us apply, the RL integral operator, ${^{R\!L}\!I^{\gamma}_s }$ , on $\tilde{H}(P_{i}, s), \; i = 1, \cdots, N+1$ ; this yields

$\begin{align} {^{R\!L}\! I}_{s}^{\gamma} \tilde{H}(P_{i}, s) & = {^{R\!L}\! I}_{s}^{\gamma}\left(\sum\limits_{k = 1}^{i}\left(i-(k-1)\right)s^{k-1}\right) = \sum\limits_{k = 1}^{i}\left(i-(k-1)\right)\left({^{R\!L}\! I}_{s}^{\gamma}s^{k-1}\right)\\ & = \sum\limits_{k = 1}^{i}\dfrac{\left(i-(k-1)\right)\Gamma(k)}{\Gamma(k+\gamma)}s^{k+\gamma-1}. \end{align}$

(2.10)

Now, using the HP, the function $s^{k+\gamma-1}$ is approximated as:

$\begin{equation} s^{k+\gamma-1}\simeq \sum\limits_{j = 1}^{N+1}e_{k, j}\tilde{H}(P_{j}, s). \end{equation}$

(2.11)

By substituting (2.11) into (2.10), we have,

$\begin{equation*} \begin{array}{lll} {^{R\!L}\! I}_{s}^{\gamma}\tilde{H}(P_{i}, s)& = \sum\limits_{k = 1}^{i}\dfrac{\left(i-(k-1)\right)\Gamma(k)}{\Gamma(k+\gamma)}\left( \sum\limits_{j = 1}^{N+1}e_{k, j}\tilde{H}(P_{j}, s)\right)\\ & = \sum\limits_{j = 1}^{N+1}\left( \sum\limits_{k = 1}^{i}\dfrac{\left(i-(k-1)\right)\Gamma(k)e_{k, j}}{\Gamma(k+\gamma)}\right)\tilde{H}(P_{j}, s)\\ & = \sum\limits_{j = 1}^{N+1}\left( \sum\limits_{k = 1}^{i}\sigma_{i, j, k}\right)\tilde{H}(P_{j}, s). \end{array} \end{equation*}$

Theorem 2.4. Suppose that $0 < \gamma\leq 1$ and $\tilde{H}(P_{i}, x)$ is the HP vector; then,

$\begin{equation*} {^{AB}\! I_{t}^{\gamma}}\mathrm{H}(s)\simeq \mathcal{I}^{\gamma}\mathrm{H}(s), \end{equation*}$

where $\mathcal{I}^{\gamma} = v_{\gamma} I+w_{\gamma}\Gamma(\gamma+1) \mathcal{P}^{\gamma}$ is called the OM of the AB-integral based on the HP and $I$ is an $(N+1)\times(N+1)$ identity matrix.

Proof. Applying the AB integral operator, ${^{AB}\! I_{s}^{\gamma}}$ , on $\mathrm{H}(s)$ yields

$\begin{equation*} {^{AB}\! I_{s}^{\gamma}}\mathrm{H}(s) = v_{\gamma} \mathrm{H}(s)+w_{\gamma}\Gamma(\gamma+1){^{RL}\! I_{s}^{\gamma}}\mathrm{H}(s). \end{equation*}$

According to Theorem 2.3, we have that ${^{R\!L}\! I_{s}^{\gamma}}\mathrm{H}(s)\simeq \mathcal{P}^{\gamma}\mathrm{H}(s)$ . Therefore

$\begin{align*} {^{AB}\! I_{s}^{\gamma}}\mathrm{H}(s)& = v_{\gamma} {H}(s)+w_{\gamma}\Gamma(\gamma+1)\mathcal{P}^{\gamma}\mathrm{H}(s)\nonumber\\ & = \left(v_{\gamma}I+w_{\gamma}\Gamma(\gamma+1)\mathcal{P}^{\gamma}\right)\mathrm{H}(s). \end{align*}$

Setting $\mathcal{I}^{\gamma} = v_{\gamma}I+w_{\gamma}\Gamma(\gamma+1)\mathcal{P}^{\gamma}$ , the proof is complete.

3. The proposed technique

The main aim of this section is to introduce a technique based on the HP of simple paths to find the solution of the TF-KEs. To do this, we first expand $D_{ss} g(s, t)$ as

$\begin{equation} D_{ss}\ g(s, t)\simeq\sum\limits_{i = 1}^{N+1}\sum\limits_{j = 1}^{N+1}h_{ij}\tilde{H}(P_{i}, s)\tilde{H}(P_{j}, t) = \mathrm{H}^{T}(s) \tilde{ h} \mathrm{H}(t). \end{equation}$

(3.1)

Integrating (3.1) with respect to $s$ gives

$\begin{equation} D_{s}g(s, t)\simeq D_{s}g(0, t)+\mathrm{H}^{T}(s)R^{T} \tilde{ h}\mathrm{H}(t). \end{equation}$

(3.2)

Again integrating the above equation with respect to $s$ gives

$\begin{equation} g(s, t)\simeq d_{1}(t)+s D_{s}g(0, t)+{H}^{T}(s)\left(R^{2}\right)^{T} \tilde{ h}{H}(t). \end{equation}$

(3.3)

By putting $s = 1$ into (3.3), we have

$\begin{equation} D_{s}g(0, t) = d_{2}(t)-d_{1}(t)- \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t). \end{equation}$

(3.4)

By substituting (3.4) into (3.3), we get

$\begin{equation} g(s, t)\simeq d_{1}(t)+s \left(d_{2}(t)-d_{1}(t)- \mathrm{H}^{T}(1)\left(R^{2}\right)^{T}\tilde{ h}{H}(t)\right)+\mathrm{H}^{T}(s)\left(R^{2}\right)^{T}\tilde{ h} \mathrm{H}(t). \end{equation}$

(3.5)

Now, we approximate that $d_{1}(t) = S_{0}^{T}\mathrm{H}(t), d_{2}(t) = S_{1}^{T}\mathrm{H}(t)$ and $s = \mathrm{H}^{T}(s)S$ and putting in (3.5), we get

$\begin{equation*} g(s, t)\simeq S_{0}^{T}\mathrm{H}(t)+\mathrm{H}^{T}(s)S \left(S_{1}^{T}\mathrm{H}(t)-S_{0}^{T}\mathrm{H}(t)- \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)\right)+\mathrm{H}^{T}(s)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t). \end{equation*}$

The above relation can be written as

$\begin{equation*} g(s, t)\simeq 1\times S_{0}^{T}\mathrm{H}(t)+\mathrm{H}^{T}(s)S \left(S_{1}^{T}\mathrm{H}(t)-S_{0}^{T}\mathrm{H}(t)- {H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)\right)+\mathrm{H}^{T}(s)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t). \end{equation*}$

Approximating $1 = \widehat{S}^{T}\mathrm{H}(s) = \mathrm{H}^{T}(s)\widehat{S}$ , the above relation is rewritten as

$\begin{align} g(s, t)&\simeq \mathrm{H}^{T}(s)\widehat{S}S_{0}^{T}\mathrm{H}(t)+{H}^{T}(s)S \left(S_{1}^{T}\mathrm{H}(t)-S_{0}^{T}\mathrm{H}(t)- \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)\right)+\mathrm{H}^{T}(x)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)\\ & = \mathrm{H}^{T}(s)\left(\widehat{S}S_{0}^{T}+S S_{1}^{T}-SS_{0}^{T}-S \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}+\left(R^{2}\right)^{T} \tilde{ h}\right)\mathrm{H}(t). \end{align}$

(3.6)

Setting $\rho_{1} = \widehat{S}S_{0}^{T}+S S_{1}^{T}-SS_{0}^{T}-S \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}+\left(R^{2}\right)^{T} \tilde{ h}$ , we have

$\begin{align} g(s, t)&\simeq \mathrm{H}^{T}(s)\rho_{1}\mathrm{H}(t). \end{align}$

(3.7)

According to (1.1), we need to obtain $D_{s}\ g(s, t)$ . Putting the approximations $d_{1}(t), d_{2}(t)$ and the relation (3.4) into (3.2) yields

$\begin{equation} D_{s}g(s, t)\simeq S_{1}^{T}\mathrm{H}(t)-S_{0}^{T}\mathrm{H}(t)- \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)+{H}^{T}(s)R^{T} \tilde{ h} \mathrm{H}(t). \end{equation}$

(3.8)

The above relation can be written as

$\begin{equation} D_{s}g(s, t)\simeq 1\times S_{1}^{T}\mathrm{H}(t)-1\times S_{0}^{T}\mathrm{H}(t)- 1\times \mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}\mathrm{H}(t)+\mathrm{H}^{T}(s)R^{T} \tilde{ h}\mathrm{H}(t). \end{equation}$

(3.9)

Putting $1 = H^{T}(s)\widehat{S}$ into the above relation, we get

$\begin{align} D_{s}g(s, t)&\simeq H^{T}(s)\widehat{S} S_{1}^{T}H(t)-H^{T}(s)\widehat{S} S_{0}^{T}H(t)- H^{T}(s)\widehat{S} H^{T}(1)\left(R^{2}\right)^{T} \tilde{ h} H(t)+H^{T}(s)R^{T} \tilde{ h} H(t)\\ & = H^{T}(s)\left(\widehat{S} S_{1}^{T}-\widehat{S} S_{0}^{T}- \widehat{S} H^{T}(1)\left(R^{2}\right)^{T}\tilde{ h}+R^{T} \tilde{ h}\right) \mathrm{H}(t). \end{align}$

(3.10)

Setting $\rho_{2} = \widehat{S} S_{1}^{T}-\widehat{S} S_{0}^{T}- \widehat{S}\mathrm{H}^{T}(1)\left(R^{2}\right)^{T} \tilde{ h}+R^{T} \tilde{ h}$ , we have

$\begin{align} D_{s}g(s, t)&\simeq \mathrm{H}^{T}(s)\rho_{2}\mathrm{H}(t). \end{align}$

(3.11)

Applying ${^{AB}\! I_{t}^{\gamma}}$ to (1.1), putting $g(s, t)\simeq \mathrm{H}^{T}(s)\rho_{1}\mathrm{H}(t), D_{s}g(s, t)\simeq \mathrm{H}^{T}(s)\rho_{2}\mathrm{H}(t)$ , $D_{ss}\ g(s, t)\simeq\mathrm{H}^{T}(s) \tilde{ h}\mathrm{H}(t)$ and approximating $\omega\left(s, t\right)\simeq\mathrm{H}^{T}(s)\rho_{3}\mathrm{H}(t)$ in (1.1) yields

$\begin{equation} \mathrm{H}^{T}(s)\rho_{1}\mathrm{H}(t) = d_{0}(s)+\vartheta_{1}(s)\mathrm {H}^{T}(s)\rho_{2}\left({^{AB}\! I_{t}^{\gamma}}\mathrm{H}(t)\right) +\vartheta_{2}(s)\mathrm{H}^{T}(s)\tilde{ h}\left({^{AB}\! I_{t}^{\gamma}}\mathrm{H}(t)\right) +{H}^{T}(s)\rho_{3}\left({^{AB}\! I_{t}^{\gamma}}\mathrm{H}(t)\right). \end{equation}$

(3.12)

Now approximating $d_{0}(s)\simeq \mathrm{H}^{T}(s)S_{2}, \vartheta_{1}(s) \simeq S_{3}^{T}\mathrm{H}(s), \vartheta_{2}(s)\simeq S_{4}^{T}\mathrm{H}(s)$ and using Theorem 2.4, the above relation can be rewritten as

$\begin{equation} {H}^{T}(s)\rho_{1}\mathrm{H}(t) = \mathrm{H}^{T}(s)S_{2}+S_{3}^{T}\mathrm{H}(s) \mathrm{H}^{T}(s)\rho_{2}\mathcal{I}^{\gamma}\mathrm{H}(t) +S_{4}^{T}\mathrm{H}(s)\mathrm{H}^{T}(s) \tilde{ h}\mathcal{I}^{\gamma}\mathrm{H}(t) +\mathrm{H}^{T}(s)\rho_{3}\mathcal{I}^{\gamma}\mathrm{H}(t). \end{equation}$

(3.13)

By Theorem 2.2, the above relation can be written as

$\begin{equation} \mathrm{H}^{T}(s)\rho_{1}\mathrm{H}(t) = \mathrm{H}^{T}(s)S_{2}\times 1+ \underset{\mathrm{H}^{T}(s)\widehat{S_{3}}}{\underbrace{S_{3}^{T}\mathrm{H}(s) {H}^{T}(s)}} \rho _{2}\mathcal{I}^{\gamma}\mathrm{H}(t) + \underset{\mathrm{H}^{T}(s)\widehat{S_{4}}}{\underbrace{S_{4}^{T}\mathrm{H}(s) {H}^{T}(s)}} \tilde{ h}\mathcal{I}^{\gamma}\mathrm{H}(t) +\mathrm{H}^{T}(s)\rho_{3}\mathcal{I}^{\gamma}\mathrm{H}(t). \end{equation}$

(3.14)

Now approximating $1 = \widehat{S}^{T}\mathrm{H}(t)$ , we have

$\begin{equation} \mathrm{H}^{T}(s)\rho_{1}\mathrm{H}(t) = \mathrm{H}^{T}(s)S_{2}\widehat{S}^{T}\mathrm{H}(t)+ \mathrm{H}^{T}(s)\widehat{S_{3}} \rho_{2}\mathcal{I}^{\gamma}\mathrm{H}(t) + \mathrm{H}^{T}(s)\widehat{S_{4}} \tilde{ h} \mathcal{I}^{\gamma}\mathrm{H}(t) +\mathrm{H}^{T}(s)\rho_{3}\mathcal{I}^{\gamma}\mathrm{H}(t). \end{equation}$

(3.15)

We can write the above relation as

$\begin{equation} \mathrm{H}^{T}(s)\left(\rho_{1}-S_{2}\widehat{S}^{T}- \widehat{S_{3}} \rho_{2}\mathcal{I}^{\gamma} - \widehat{S_{4}} \tilde{ h} \mathcal{I}^{\gamma} -\rho_{3}\mathcal{I}^{\gamma}\right)\mathrm{H}(t) = 0. \end{equation}$

(3.16)

Therefore we have

$\begin{equation} \rho_{1}-S_{2}\widehat{S}^{T}- \widehat{S_{3}} \rho_{2}\mathcal{I}^{\gamma} - \widehat{S_{4}} \tilde{ h}\mathcal{I}^{\gamma} -\rho_{3}\mathcal{I}^{\gamma} = 0. \end{equation}$

(3.17)

By solving the obtained system, we find $h_{ij}$ , $i, j = 1, 2, \cdots, N+1$ . Consequently, $g(s, t)$ can be calculated by using (3.7).

4. Convergence analysis

Set $I = (a, b)^{n}, \, n = 2, 3$ in $\mathbb{R}^{n}$ . The Sobolev norm is given as

$\begin{equation*} \|g\|_{H^\epsilon(I)} = \left(\sum\limits_{k = 0}^\epsilon\sum\limits_{l = 0}^n \|D_{l}^{(k)} g\|^{2}_{L^2(I)}\right)^{\frac{1}{2}}, \; \; \; \; \epsilon\geq 1, \end{equation*}$

where $D_{l}^{(k)}u$ and $H^\epsilon (I)$ are the $k$ -th derivative of $g$ and Sobolev space, respectively. The notation $\vert g\vert_{H^{\epsilon; N}}$ is given as ^[35]

$\begin{equation*} |g|_{H^{\epsilon;N}(I)} = \left(\sum\limits_{k = \min\{\epsilon, N+1\}}^\epsilon\sum\limits_{l = 0}^n \|D_{l}^{(k)} g\|^{2}_{L^2(I)}\right)^{\frac{1}{2}}. \end{equation*}$

Theorem 4.1 (See ^[36]). Let $g(s, t)\in H^{\epsilon}(I)$ with $\epsilon\geq 1$ . Considering $P_{N}g(s, t) = \sum\limits_{r = 1}^{N+1} \sum\limits_{n = 1}^{N+1}a_{r, n}P_{r}(s) P_{n}(t)$ as the best approximation of $g(s, t)$ , we have

$\begin{equation*} \|g-P_{N}g\|_{L^2(I)}\leq CN^{1-\epsilon}|g|_{H^{\epsilon;N}(I)}, \end{equation*}$

and if $1\leq \iota\leq \epsilon$ , then

$\begin{equation*} \|g-P_{N}g\|_{H^{\iota}(I)}\leq CN^{\vartheta(\iota)-\epsilon}|g|_{H^{\epsilon;N}(I)}, \end{equation*}$

with

$\begin{equation*} \vartheta(\iota) = \left\{ \begin{array}{ll} 0, &\iota = 0, \\ 2\iota-\frac{1}{2}, &\iota > 0. \end{array} \right. \end{equation*}$

Lemma 4.1. The AB derivative can be written by using the fractional order RL integral as follows:

$\begin{equation*} {^{AB}\! D}_{t}^{\gamma}g(t) = \dfrac{\Phi(\gamma)}{1-\gamma}\sum\limits_{l = 0}^{\infty}\varpi^{l}\, {^{R\!L}\!I^{l\gamma+1}_t }g'(t), \; \; \; \; \varpi = -\frac{\gamma}{1-\gamma}. \end{equation*}$

Proof. According to the definitions of the AB derivative and the RL integral, the proof is complete.

Theorem 4.2. Suppose that $0 < \gamma\leq 1, \, \left|\vartheta_{1}(s)\right|\leq \tau_{1}, \, \left|\vartheta_{2}(s)\right|\leq \tau_{2}$ and $g(s, t)\in H^{\epsilon}(I)$ with $\epsilon\geq 1$ . If $E(s, t)$ is the residual error by approximating $g(s, t)$ , then $E(s, t)$ can be evaluated as

$\begin{equation*} \|E(s, t)\|_{L^{2}(I)}\leq \varrho_{1}\left(|g|^*_{H^{\epsilon;N}(I)}+|\partial_{s}g|^*_{H^{\epsilon;N}(I)}\right), \end{equation*}$

where $1\leq \iota\leq \epsilon$ and $\varrho_{1}$ is a constant number.

Proof. According to (1.1),

$\begin{equation} {^{AB}\! D}_{t}^{\gamma}\ g(s, t) = \vartheta_{1}(s)D_{s}\ g(s, t)+\vartheta_{2}(s)D_{ss}\ g(s, t)+\omega\left(s, t\right), \end{equation}$

(4.1)

and

$\begin{equation} {^{AB}\! D}_{t}^{\gamma}\ g_{N}(s, t) = \vartheta_{1}(s)D_{s}\ g_{N}(s, t)+\vartheta_{2}(s)D_{ss}\ g_{N}(s, t)+\omega\left(s, t\right). \end{equation}$

(4.2)

Substituting Eqs (4.1) and (4.2) in $E(s, t)$ yields

$\begin{equation*} E(s, t) = {^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))+\vartheta_{1}(s) D_{s}(g_{N}(s, t)-g(s, t))+\vartheta_{2}(s) D_{ss}(g_{N}(s, t)-g(s, t)). \end{equation*}$

and then

$\begin{equation} \begin{split} \|E(s, t)\|^{2}_{L^{2}(I)}\leq&\|{^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))\|^{2}_{L^{2}(I)}\\ &+\tau_{1}\, \| D_{s}(g(s, t)-g_{N}(s, t))\|^{2}_{L^{2}(I)}\\ &+\tau_{2} \, \|D_{ss}(g(s, t)-g_{N}(s, t))\|^{2}_{L^{2}(I)}.\\ \end{split} \end{equation}$

(4.3)

Now, we must find a bound for $\|{^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))\|_{L^{2}(I)}$ . In view of ^[26], and by using Lemma 4.1, in a similar way, we write

$\begin{equation*} \begin{array}{ll} \|{^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))\|^{2}_{L^{2}(I)}& = \|\frac{\Phi(\gamma)}{1-\gamma} \sum\limits_{l = 0}^{\infty}\varpi^{l}\, {^{R\!L}\!I^{l\gamma+1}_t }(D_{t}g(s, t)-D_{t}g_{N}(s, t))\|^{2}_{L^{2}(I)}\\ &\leq\left(\frac{\Phi(\gamma)}{1-\gamma} \sum\limits_{l = 0}^{\infty}\frac{\varpi^{l}}{\Gamma(l \gamma+2)}\right)^{2}\|D_{t}g(s, t)-D_{t}g_{N}(s, t)\|^{2}_{L^{2}(I)}\\ &\leq \left(\frac{\Phi(\gamma)}{1-\gamma}E_{\gamma, 2}(\varpi)\right)^{2}\|g(s, t)-g_{N}(s, t)\|^{2}_{H^{\iota}(I)}. \end{array} \end{equation*}$

Therefore,

$\begin{equation} \|{^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))\|_{L^{2}(I)}\leq \delta_{1} CN^{\vartheta(\iota)-\epsilon}|g|_{H^{\epsilon;N}(I)}, \end{equation}$

(4.4)

where $\frac{\Phi(\gamma)}{1-\gamma}E_{\gamma, 2}(\varpi)\leq \delta_{1}$ . Thus, from (4.4), we can write

$\begin{equation} \|{^{AB}\! D}_{t}^{\gamma}(g(s, t)-g_{N}(s, t))\|^{2}_{L^{2}(I)}\leq \delta_{1}|g|^*_{H^{\epsilon;N}(I)}, \end{equation}$

(4.5)

where $|g|^*_{H^{\epsilon; N}(I)} = C N^{\vartheta(\iota)-\epsilon}|g|_{H^{\epsilon; N}(I)}$ . By Theorem 4.1,

$\begin{equation} \| D_{s}(g(s, t)-g_N(s, t))\|_{L^{2}(I)}\leq CN^{\vartheta(\iota)-\epsilon}|g|_{H^{\epsilon;N}(I)} = |g|^*_{H^{\epsilon;N}(I)}, \end{equation}$

(4.6)

and

$\begin{equation} \begin{array}{ll} \|D_{ss}(g(s, t)-g_{N}(s, t))\|_{L^{2}(I)} & = \|D_{s}\left(D_{s}(g(s, t)- g_{N}(s, t))\right)\|_{L^{2}(I)}\\ & \leq\|D_{s}g(s, t)-D_{s}g_{N}(s, t)\|_{H^{\iota}(I)}\\ &\leq |D_{s}g|^*_{H^{\epsilon;N}(I)}, \end{array} \end{equation}$

(4.7)

where $|D_{s}g|^*_{H^{\epsilon; N}(I)} = C N^{\vartheta(\iota)-\epsilon}|D_{s}g|_{H^{\epsilon; N}(I)}$ . Taking $\varrho_{1} = \max\{\delta_{1}+ \tau_{1}, \tau_{2}\}$ and substituting (4.5)–(4.7) into (4.3); then, the desired result is obtained.

5. Test problems

In this section, the proposed technique which is described in Section 3 is shown to be tested using some numerical examples. The codes are written in Mathematica software.

Example 5.1. Consider (1.1) with $\vartheta_{1}(s) = -1, \vartheta_{2}(s) = 0.1$ and $\omega\left(s, t\right) = 0$ . The initial and boundary conditions can be extracted from the analytical solution $g(s, t) = \tau_{0}e^{\tau_{1}t-\tau_{2}s}$ when $\gamma = 1$ . Setting $\tau_{0} = 1, \tau_{1} = 0.2, \tau_{2} = \dfrac{\vartheta_{1}(s)+\sqrt{\vartheta_{1}^{2}(s)+4\vartheta_{2}(s)\tau_1}}{2\vartheta_{2}(s)}$ , considering $N = 3$ and using the proposed technique, the numerical results of the TF-ADE are reported in Tables 1 and 2, and in Figures 1–3.

Table 1. (Example 5.1) Numerical results of the absolute error when

$\gamma = 0.99, N = 3$ ,

$t = 1$ .

$s$	Method of ^[21]	The presented method
$0.1$	$1.05799e-2$	$3.86477e-4$
$0.2$	$1.21467e-2$	$1.33870e-4$
$0.3$	$4.94776e-3$	$4.08507e-5$
$0.4$	$2.35280e-4$	$1.48842e-4$
$0.5$	$2.36604e-3$	$2.01089e-4$
$0.6$	$1.08676e-2$	$2.08410e-4$
$0.7$	$2.18851e-2$	$1.81459e-4$
$0.8$	$2.91950e-2$	$1.30730e-4$
$0.9$	$2.49148e-2$	$6.65580e-5$

| Show Table

DownLoad: CSV

Table 2. (Example 5.1) Numerical results of the absolute error when

$\gamma = 0.99, N = 3$ ,

$s = 0.75$ .

$t$	Method of ^[19]	The presented method
$0.1$	$1.13874e-3$	$2.15272e-3$
$0.2$	$1.41664e-3$	$2.32350e-3$
$0.3$	$1.62234e-3$	$2.30934e-3$
$0.4$	$1.76917e-3$	$2.14768e-3$
$0.5$	$1.87045e-3$	$1.87583e-3$
$0.6$	$1.93953e-3$	$1.53092e-3$
$0.7$	$1.98971e-3$	$1.14997e-3$
$0.8$	$2.03434e-3$	$7.69801e-4$
$0.9$	$2.08671e-3$	$4.27112e-4$

| Show Table

DownLoad: CSV

Figure 1. (Example 5.1) The absolute error at some selected points when (a)

$\gamma = 0.8$ , (b)

$\gamma = 0.9$ , (c)

$\gamma = 0.99$ , (d)

$\gamma = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. (Example 5.1) Error contour plots when (a)

$\gamma = 0.99$ , (b)

$\gamma = 1$ , (c)

$\gamma = 0.8$ , (d)

$\gamma = 0.9$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. (Example 5.1) The absolute error at some selected points when (a)

$\gamma = 0.8$ , (b)

$\gamma = 0.9$ , (c)

$\gamma = 0.99$ , (d)

$\gamma = 1$ .

DownLoad: Full-Size Img PowerPoint

Example 5.2. Consider (1.1) with $\vartheta_{1}(s) = s, \vartheta_{2}(s) = \dfrac{s^{2}}{2}$ and $\omega\left(s, t\right) = 0$ . The initial and boundary conditions can be extracted from the analytical solution $g(s, t) = s E_{\alpha}(t^{\alpha})$ . By setting $N = 5$ and using the proposed technique, the numerical results of the TF–KE are as reported in Figures 4–6.

Figure 4. (Example 5.2) The absolute error at some selected points when (a)

$\gamma = 0.7$ , (b)

$\gamma = 0.8$ , (c)

$\gamma = 0.9$ , (d)

$\gamma = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. (Example 5.2) Error contour plots when (a)

$\gamma = 0.7$ , (b)

$\gamma = 0.8$ , (c)

$\gamma = 0.9$ , (d)

$\gamma = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. (Example 5.2) The absolute error at some selected points when (a)

$\gamma = 0.7$ , (b)

$\gamma = 0.8$ , (c)

$\gamma = 0.9$ , (d)

$\gamma = 1$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

Time fractional Kolmogorov equations and time fractional advection-diffusion equations have been used to model many problems in mathematical physics and many scientific applications. Developing efficient methods for solving such equations plays an important role. In this paper, a proposed technique is used to solve TF-ADEs and TF-KEs. This technique reduces the problems under study to a set of algebraic equations. Then, solving the system of equations will give the numerical solution. An error estimate is provided. This method was tested on a few examples of TF-ADEs and TF-KEs to check the accuracy and applicability. This method might be applied for system of fractional order integro-differential equations and partial differential equations as well.

Use of AI tools declaration

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

Acknowledgments

The authors would like to thank for the support from Scientific Research Fund Project of Yunnan Provincial Department of Education, No. 2022J0949. The authors also would like to thank the anonymous reviewers for their valuable and constructive comments to improve our paper.

Conflict of interest

The authors declare there is no conflicts of interest.

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