Transient prophylaxis and multiple epidemic waves

Rebecca C. Tyson; Noah D. Marshall; Bert O. Baumgaertner; Rebecca C. Tyson; Noah D. Marshall; Bert O. Baumgaertner

doi:10.3934/math.2022311

AIMS Mathematics

2022, Volume 7, Issue 4: 5616-5633. doi: 10.3934/math.2022311

Previous Article Next Article

Research article Special Issues

Transient prophylaxis and multiple epidemic waves

1.
CMPS Department (Mathematics), University of British Columbia Okanagan, Kelowna, BC, Canada
2.
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada
3.
Department of Politics and Philosophy, University of Idaho, Moscow, Idaho, USA

Received: 06 January 2021 Revised: 21 May 2021 Accepted: 06 October 2021 Published: 10 January 2022
MSC : 91D10, 92D30

Public opinion and opinion dynamics can have a strong effect on the transmission rate of an infectious disease for which there is no vaccine. The coupling of disease and opinion dynamics however, creates a dynamical system that is complex and poorly understood. We present a simple model in which susceptible groups adopt or give up prophylactic behaviour in accordance with the influence related to pro- and con-prophylactic communication. This influence varies with disease prevalence. We observe how the speed of the opinion dynamics affects the total size and peak size of the epidemic. We find that more reactive populations will experience a lower peak epidemic size, but possibly a larger final size and more epidemic waves, and that an increase in polarization results in a larger epidemic.

Keywords:

Citation: Rebecca C. Tyson, Noah D. Marshall, Bert O. Baumgaertner. Transient prophylaxis and multiple epidemic waves[J]. AIMS Mathematics, 2022, 7(4): 5616-5633. doi: 10.3934/math.2022311

Related Papers:

[1]	Mahmoud A. Zaky, Weam G. Alharbi, Marwa M. Alzubaidi, R.T. Matoog . A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations. AIMS Mathematics, 2025, 10(3): 7067-7085. doi: 10.3934/math.2025322
[2]	Khadijeh Sadri, David Amilo, Kamyar Hosseini, Evren Hinçal, Aly R. Seadawy . A tau-Gegenbauer spectral approach for systems of fractional integro-differential equations with the error analysis. AIMS Mathematics, 2024, 9(2): 3850-3880. doi: 10.3934/math.2024190
[3]	Obaid Algahtani, M. A. Abdelkawy, António M. Lopes . A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations. AIMS Mathematics, 2022, 7(8): 15453-15470. doi: 10.3934/math.2022846
[4]	Chuanli Wang, Biyun Chen . An $hp$ -version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels. AIMS Mathematics, 2023, 8(8): 19816-19841. doi: 10.3934/math.20231010
[5]	Abeer Al Elaiw, Farva Hafeez, Mdi Begum Jeelani, Muath Awadalla, Kinda Abuasbeh . Existence and uniqueness results for mixed derivative involving fractional operators. AIMS Mathematics, 2023, 8(3): 7377-7393. doi: 10.3934/math.2023371
[6]	Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi . A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151
[7]	Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad, Mohammed S. Abdo, Kamal Shah, Thabet Abdeljawad . Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative. AIMS Mathematics, 2022, 7(9): 15994-16016. doi: 10.3934/math.2022876
[8]	Ahmed M. Rajab, Saeed Pishbin, Javad Shokri . Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives. AIMS Mathematics, 2024, 9(9): 23182-23196. doi: 10.3934/math.20241127
[9]	Zhengang Zhao, Yunying Zheng, Xianglin Zeng . Finite element approximation of fractional hyperbolic integro-differential equation. AIMS Mathematics, 2022, 7(8): 15348-15369. doi: 10.3934/math.2022841
[10]	Amnah E. Shammaky, Eslam M. Youssef . Analytical and numerical techniques for solving a fractional integro-differential equation in complex space. AIMS Mathematics, 2024, 9(11): 32138-32156. doi: 10.3934/math.20241543

Abstract

1. Introduction

High-order pantograph Volterra integro-differential equations provide a vital framework for describing a number of natural phenomena in physics and engineering ^[1,2,3]. With the integral form, the pantograph Volterra integro-differential equation has the ability to model systems with memory influence, and the pantograph term adds to the equation's capacity to study scaling properties. These attributes ensure the appearance of pantograph Volterra integro-differential equations in diverse fields, including wave propagation and quantum mechanics ^[4,5,6]. It has an essential value in studying systems where the scaled spatial or temporal variables and past states help in obtaining the behavior of the wave, for instance, quantum transitions, scattering processes, wave propagation in non-classical structures, and optical interference and diffraction ^[7,8].

On the other hand, the pantograph term as well as the differential and integral terms make it uneasy to find the exact solution of pantograph Volterra integro-differential equations, so many authors are interested in finding valid and accurate analytical and numerical solutions for different kinds of such equations. Abbaszadeh et al. ^[9] developed a virtual element framework for nonlinear partial integro-differential equations, proposing two temporal strategies (uniform and graded meshes) to address integral term singularity establishing a fully discrete scheme with rigorous error estimates, unconditional stability, convergence proofs, and validation through numerical experiments on varied physical domains. Zaky et al. ^[10] proposed a spectral tau approach to approximate the solution of high-order pantograph Volterra-Fredholm integro-differential equations in both one- and two-dimensions. Abdelhakem ^[11] utilized the Pseudo-spectral integration matrices for fractional Volterra integro-differential equations and Abel's integral equations. Ghoreyshi et al. ^[12] investigated a nonlinear time-fractional partial integro-differential equation by combining the weighted and shifted Grünwald–Letnikov formula for temporal discretization of the Caputo fractional derivative, the fractional trapezoidal rule for the Volterra integral, and Chebyshev spectral-collocation methods for spatial approximation. Ghosh and Mohapatra ^[13] applied an iterative technique for nonlinear delay Volterra integro-differential equations, where the integral term is approximated using the composite trapezoidal rule, and the Daftardar-Gejji and Jafari technique is used to solve the resulting implicit algebraic equation. Alsuyuti et al. ^[14] extended the application of the Galerkin spectral method for pantograph integro-differential and systems of pantograph differential equations of arbitrary order. Behera and Ray ^[15] carried out an operational scheme based on Bernoulli and müntz–Legendre wavelets (BMLW) for the solution of pantograph Volterra delay-integro-differential equations. Elkot et al. ^[16] presented the multi-variate Legendre-collocation spectral scheme for multi-dimensional nonlinear Volterra–Fredholm integral equations. Zhao et al. ^[17] utilized the Lagrange interpolation (LIT) and Bernstein tau (BT) methods, respectively, to deal with linear pantograph Volterra delay-integro-differential equations. The author in ^[18] applied the Jacobi spectral tau approach for systems of multi-pantograph equations. Bica and Satmari ^[19] employed an iterative approach based on the Bernstein composite quadrature and fuzzy Bernstein spline interpolation methods for the approximate solution of pantograph fuzzy Volterra integral equations. In ^[20], the spectral collocation technique is implemented, where the integral terms are approximated via the Gauss quadrature rule, for systems of multi-dimensional integral equations. In ^[21], the multistep collocation method is carried out for the pantograph second-kind Volterra integral equation. Other numerical methods include ^[22,23].

This manuscript aims to provide a numerical solution for the high-order multi-pantograph Volterra integro-differential equation

$\begin{equation} \begin{split} & \sum\limits_{i = 0}^{s} f^{(i)}(x) = g(x)+\sum\limits_{i = 0}^{s}\mu_i(x) f^{(i)}(c_i x)+\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}\nu_i(x) f^{(i)}(w)dw + \int_{0}^{x}\xi(w) f(w)dw, \\ &f^{(i)}(0) = \theta_i, \qquad i = 0, 1, \cdots, s-1, \qquad 0\leq x\leq\alpha, \end{split} \end{equation}$

(1.1)

where $\xi(x), \ g(x), \ \mu_i(x), \ \nu_i(x)$ and $\gamma_i(x)\ (0 < i < s)$ are known functions and $\theta_i, \ c_i$ and $d_i\ (0 < i < s)$ are real numbers with $0 < c_i, d_i < 1$ . In this regard, we use the operational matrix of differentiation and derive another one of pantograph, based on shifted Jacobi polynomials, that are utilized in conjunction with the spectral collocation approach to simplify the problem to an equivalent one of solving a system of algebraic equations. In addition, we study the application of the considered numerical technique to the system of high-order multi-pantograph Volterra integro-differential equations. On the other hand, we aim to extend the constructed numerical technique to solve the two-dimensional high-order multi-pantograph Volterra integro-differential equation

$\begin{equation} \begin{cases} \begin{split} \frac{\partial^{s+q} f(x, y) }{\partial x^{s}\partial y^{q}} = &g(x, y)+ \sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\int_{0}^{c_i x}\int_{0}^{d_j y} \xi_{i, j}(t, u) \frac{\partial^{i+j} f(t, u)}{\partial t^{i}\partial u^{j}} dtdu\\ &+ \int_{0}^{y}\int_{0}^{x}\nu(t, u) f(t, u) dtdu + \sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\mu_{i, j}(x, y)\frac{\partial^{i+j} f(c_ix, d_jy)}{\partial x^{i}\partial y^{j}}, \end{split}\\ \frac{\partial^{i} f(0, y) }{\partial x^{i}} = \theta_{i}(y), \qquad\quad i = 0, 1, \cdots, s-1, \quad 0\leq y \leq \beta, \\ \frac{\partial^{j} f(x, 0) }{\partial y^{j}} = \vartheta_{j}(x), \qquad\quad j = 0, 1, \cdots, q-1, \quad 0\leq x \leq \alpha, \end{cases} \end{equation}$

(1.2)

where $\mu_{i, j}(x, y), \ \xi_{i, j}(x, y), \ \nu(x, y), \ \theta_i(y)$ and $\vartheta_j(x)\ (0 < i < s, \ 0 < j < q)$ are known functions with $0 < c_i, d_j < 1.$

The paper is outlined as follows: Section 2 investigates the existence and uniqueness of the high-order multi-pantograph Volterra integro-differential equation (1.1). Section 3 provides some important properties of shifted Jacobi polynomials. Section 4 presents the application of operational matrices of pantograph and differentiation in conjunction with the Jacobi spectral collocation method to solve high-order multi-pantograph Volterra integro-differential equations. Section 5 studies the application of the considered numerical approach to the system of high-order multi-pantograph Volterra integro-differential equations. Section 6 discusses the extension of the numerical approach discussed in the previous section to solve the two-dimensional high-order multi-pantograph Volterra integro-differential equation. Section 7 introduces several test problems with their numerical solutions and comparisons with other spectral methods in the literature. Concluding remarks are displayed in Section 8.

2. Existence and uniqueness

In this section we study the existence and uniqueness of the solution of the high-order multi-pantograph Volterra integro-differential equation (1.1).

Theorem 1. (^[24] Gronwall inequality) Let $f(x)$ be a non-negative integrable function over $(0, \alpha]$ , and let $g(x)$ and $E(x)$ be continuous functions on $[0, \alpha]$ . If $E(x)$ satisfies

$\begin{equation} E(x) \leq g(x) + \int_{0}^{x} f(w) E(w) dw, \qquad \forall x \in [0, \alpha], \end{equation}$

(2.1)

then we have

$\begin{equation} E(x) \leq g(x) + \int_{0}^{x} f(t) g(t) exp\left(\int_{t}^{x}f(w)dw\right) dt, \qquad \forall x \in [0, \alpha]. \end{equation}$

(2.2)

If $g(x)$ is non-decreasing, then

$\begin{equation} E(x) \leq g(x) exp\left(\int_{0}^{x}f(w)dw\right) dt, \qquad \forall x \in [0, \alpha]. \end{equation}$

(2.3)

Theorem 2. Assume $f(x)$ is a continuous function in $[0, \alpha];$ then the pantograph Volterra integro-differential equation (1.1) has a unique solution.

Proof. Firstly, we transform high-order multi-pantograph Volterra integro-differential equation (1.1) into its equivalent integral form as follows:

$\begin{equation} f(x)-\sum\limits_{i = 0}^{s-1}\theta_i \frac{x^i}{i!} = \overbrace{\int_{0}^{x}\int_{0}^{x}\cdots\int_{0}^{x}}^{s\ \text{times}}\mathcal{K}\left(x, f(x)\right)\overbrace{dx\cdots dxdx}^{s\ \text{times}}, \end{equation}$

(2.4)

where

$\mathcal{K}\left(x, f(x)\right) = g(x)+\sum\limits_{i = 0}^{s}\mu_i(x) f^{(i)}(c_i x)+\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}\nu_i(w) f^{(i)}(w)dw + \int_{0}^{x}\xi(w) f(w)dw+\sum\limits_{i = 0}^{s-1} \gamma_i(x) f^{(i)}(x).$

Suppose that for any continuous functions $f(x), \ h(x), \ \xi(x), \ \mu_i(x), \ \gamma_i(x)$ and $\nu_i(x)\ (0\leq i \leq s),$ in $[0, \alpha],$ and $c_i\ (0\leq i \leq s),$ with $0 < c_i, d_i < 1,$ the following inequalities hold:

$\begin{equation} \begin{split} &\left\| f^{(i)}(x)-h^{(i)}(x)\right\|_\infty \leq A \left\| f(x)-h(x)\right\|_\infty, \qquad \qquad \quad \forall 0\leq i \leq s, \\ &\left\| f(c_ix)-h(c_ix)\right\|_\infty \leq B \left\| f(x)-h(x)\right\|_\infty, \qquad \qquad \quad\ \forall 0\leq i \leq s, \\ &\left\| \xi(x) \right\|_\infty \leq C, \quad \left\| \mu_i(x)\right\|_\infty\leq D, \quad \left\|\nu_i(x)\right\|_\infty\leq E, \quad \left\|\gamma_i(x)\right\|_\infty\leq F, \qquad \forall 0\leq i \leq s.\\ \end{split} \end{equation}$

(2.5)

Now, we prove that $\mathcal{K}\left(x, f(x)\right)$ satisfies the Lipschitz condition. For any two continuous functions $f(x), \ h(x),$ we have

$\begin{split} \big\|\mathcal{K}\left(x, f(x)\right)-\mathcal{K}\left(x, h(x)\right)\big\|_\infty& = \Bigg\|\sum\limits_{i = 0}^{s}\mu_i(x) \left(f^{(i)}(c_i x)-h^{(i)}(c_i x)\right)+\sum\limits_{i = 0}^{s-1}\gamma_i(x) \left(f^{(i)}(x)-h^{(i)}(x)\right)\\ +\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}&\nu_i(w) \left(f^{(i)}(w)-h^{(i)}(w)\right)dw + \int_{0}^{x}\xi(w) \left(f(w)-h(w)\right) dw\Bigg\|_\infty\\ &\leq \sum\limits_{i = 0}^{s}|\mu_i(x)| \left\|f^{(i)}(c_i x)-h^{(i)}(c_i x)\right\|_\infty+ \sum\limits_{i = 0}^{s-1}|\gamma_i(x)| \left\|f^{(i)}(x)-h^{(i)}(x)\right\|_\infty\\ +\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}&|\nu_i(w)| \left\|f^{(i)}(w)-h^{(i)}(w)\right\|_\infty dw + \int_{0}^{x}|\xi(w)| \left\|f(w)-h(w)\right\|_\infty dw\\ &\leq \mathcal{L}\left\|f(x)-h(x)\right\|_\infty, \\ \end{split}$

where $\mathcal{L} = ABD+ AE\alpha+ C\alpha+AF$ is a positive constants. Consequently, Eq (2.4) can be rewritten as

$\begin{equation} f(x)-\sum\limits_{i = 0}^{s-1}\theta_i \frac{x^i}{i!} = \frac{1}{(s-1)!}\int_{0}^{x} (x-t)^{s-1}\mathcal{K}\left(t, f(t)\right)dt. \end{equation}$

(2.6)

Assume there are two solutions $f(x)$ and $h(x)$ of the pantograph Volterra integro-differential equation (1.1) satisfying $f^{(i)}(0) = h^{(i)}(0) = \theta_i, \ i = 0, 1, \cdots, s.$ This, along with Eq (2.6), yields

$\begin{equation} \begin{split} \left\| f(x)-h(x)\right\|_\infty & = \left\|\frac{1}{(s-1)!}\int_{0}^{x} (x-t)^{s-1}\left(\mathcal{K}\left(t, f(t)\right)-\mathcal{K}\left(t, h(t)\right)\right)dt \right\|_\infty\\ &\leq\frac{1}{(s-1)!}\int_{0}^{x} \left|(x-t)^{s-1}\right| \left\|\mathcal{K}\left(t, f(t)\right)-\mathcal{K}\left(t, h(t)\right)\right\|_\infty dt \\ &\leq\frac{\mathcal{L}}{(s-1)!}\int_{0}^{x} \left|(x-t)^{s-1}\right|\left\| f(x)-h(x)\right\|_\infty dt, \end{split} \end{equation}$

(2.7)

and then, it follows from Theorem 1 that $\left\| f(x)-h(x)\right\|_\infty = 0.$ Now, we consider

$\begin{equation} f_l(x)-\sum\limits_{i = 0}^{s-1}\theta_i \frac{x^i}{i!} = \frac{1}{(s-1)!}\int_{0}^{x} (x-t)^{s-1}\mathcal{K}\left(t, f_{l-1}(t)\right)dt, \end{equation}$

(2.8)

and define $\mathfrak{E}_{l+1} = f_{l+1}(x)-f_{l}(x).$ Following the proof in (^[25], Section 3), we get

$\begin{equation} \left\| \mathfrak{E}_{l+1}\right\|_\infty = \left\|\frac{1}{(s-1)!}\int_{0}^{x} (x-t)^{s-1}\left(\mathcal{K}\left(t, f_{l+1}(t)\right)-\mathcal{K}\left(t, f_l(t)\right)\right)dt \right\|_\infty \leq \mathcal{L} \left\| \mathfrak{E}_{l}\right\|_\infty\rightarrow 0, \; \mathcal{L} < 1 . \end{equation}$

(2.9)

Thus, for $n, m \rightarrow 0,$ we have

$\begin{equation} \left\| f_n(x)-f_m(x)\right\|_\infty \leq \left\| \mathfrak{E}_n\right\|_\infty+\left\| \mathfrak{E}_{n-1}\right\|_\infty+\cdots+\left\| \mathfrak{E}_{m-1}\right\|_\infty\rightarrow 0, \; n > m, \end{equation}$

(2.10)

which implies that there is only one solution of (1.1). □

3. Shifted Jacobi polynomials

Denote $S_{ \alpha, l}^{(a, b)}(x),$ for $l = 0, 1, \cdots,$ by the shifted Jacobi polynomials defined in $\Lambda_x = [0, \alpha]$ , then it can be given analytically by

$S_{ \alpha, l}^{(a, b)}(x) = \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} \, x^i = \sum\limits_{i = 0}^{l} \dfrac{(-1)^{l-i}\Gamma(l+b+1)\Gamma(l+i+a+b+1)}{\Gamma(i+b+1)\Gamma(l+a+b+1)l!(l-i)! \alpha^i} x^i,$

and satisfy

$\frac{d^q}{dx^q} S_{\alpha, l}^{(a, b)}(0) = (-1)^{l-q} \frac{\Gamma(l +b + 1) (l + a+ b +1)}{\Gamma(l - q+1) \Gamma(q+b+1) \alpha^q}.$

Shifted Jacobi polynomials satisfy the orthogonality relation

$\int_0^\alpha S_{\alpha, l}^{(a, b)}(x) S_{\alpha, m}^{(a, b)}(x) w_\alpha^{(a, b)} dx = h_{\alpha, l}^{(a, b)} \delta_{lm},$

where $w_ \alpha^{(a, b)} = x^b (\alpha-x)^a$ , $h_{ \alpha, l}^{(a, b)} = \dfrac{\alpha^{a+b+1} \Gamma(l+a+1) \Gamma(l+b+1)}{(2l+a+b+1) l! \Gamma(l+a+b+1)}$ and $\delta_{lm}$ is the well-known Kronecker delta function.

Any function $f\in L^2(\Lambda_x)$ can be expanded based on shifted Jacobi polynomials by:

$\begin{equation} f(x) = \sum\limits_{l = 0}^{\infty}f_l S_{ \alpha, l}^{(a, b)}(x); \qquad f_l = \frac{1}{h_{ \alpha, l}^{(a, b)}} \int_{0}^{ \alpha} f(x) S_{ \alpha, l}^{(a, b)}(x) w_ \alpha^{(a, b)} dx. \end{equation}$

(3.1)

The shifted Jacobi-Gauss quadrature rule satisfies

$\begin{equation} \int_0^\alpha f(x) w^{(a, b)}_\alpha(x) \, dx = \frac{\alpha}{2} \sum\limits_{j = 0}^{\mathcal{L}} w_{\mathcal{L}, j}^{(a, b)} f\left(\frac{\alpha}{2} \left(1 + x^{(a, b)}_{\mathcal{L}, j}\right)\right), \end{equation}$

(3.2)

where $\left\{x^{(a, b)}_{\mathcal{L}, j}\right\}_{j = 0}^{\mathcal{L}}$ are the zeros of $S_{\mathcal{L}+1}^{(a, b)}(x)$ , and $\left\{w^{(a, b)}_{\mathcal{L}, j}\right\}_{j = 0}^{\mathcal{L}}$ are their corresponding weights.

If we denote $P^{ \alpha}_{\mathcal{L}}$ by the orthogonal projection:

$P^{ \alpha}_{\mathcal{L}}: L^2(\Lambda)\rightarrow \mathcal{S}_{\mathcal{L}}^ \alpha; \qquad \mathcal{S}_{\mathcal{L}}^ \alpha = \text{Span}\{S_{ \alpha, l}^{(a, b)}(x): \qquad 0 \leq l \leq \mathcal{L}\},$

then, we have

$\begin{equation} P^{ \alpha}_{\mathcal{L}}f = f_{\mathcal{L}}(x) = \sum\limits_{l = 0}^{\mathcal{L}} f_l S_{ \alpha, l}^{(a, b)}(x) = \mathcal{F}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{equation}$

(3.3)

where

$\begin{equation} \mathcal{F}_\mathcal{L} = [f_l, \quad 0 \leq l \leq \mathcal{L}]^T, \qquad \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x) = [ S_{ \alpha, l}^{a, b}(x), \quad 0 \leq l \leq \mathcal{L}]^T. \end{equation}$

(3.4)

Similarly, for $f\in L^2(\Lambda_x\times \Lambda_y), \ \Lambda_x = [0, \alpha], \ \Lambda_y = [0, \beta],$ we have

$\begin{equation} f_{\mathcal{L}, \mathcal{M}}(x, y) = \sum\limits_{l = 0}^{\mathcal{L}} \sum\limits_{m = 0}^{\mathcal{M}} f_{l, m} S_{ \alpha, l}^{a, b}(x) S_{ \beta, m}^{a, b}(y) = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{equation}$

(3.5)

where $\mathcal{F}^T_{\mathcal{L}, \mathcal{M}}$ and $\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y)$ are defined by:

$\begin{equation} \begin{split} \mathcal{F}^T_{\mathcal{L}, \mathcal{M}}& = [f_{l, m}, \qquad 0 \leq l \leq \mathcal{L}, \ 0 \leq m \leq \mathcal{M}]^T, \\ \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y)& = [S_{ \alpha, l}^{a, b}(x)S_{ \beta, m}^{a, b}(y), \qquad 0 \leq l \leq \mathcal{L}, \ 0 \leq m \leq \mathcal{M}]^T, \end{split} \end{equation}$

(3.6)

and

$\begin{equation} f_{l, m} = \frac{1}{ h_{ \alpha, l}^{(a, b)} h_{ \beta, m}^{(a, b)}}\int_{0}^{ \alpha}\int_{0}^{ \beta} f(x, y) S_{ \alpha, l}^{a, b}(x)S_{ \beta, m}^{a, b}(y) w_{ \alpha}^{(a, b)}(x)w_{ \beta}^{(a, b)}(y) dxdy. \end{equation}$

(3.7)

4. Multi-pantograph Volterra integro-differential equations

In this section, we apply the Jacobi spectral collocation method to solve the following high-order multi-pantograph Volterra integro-differential equation (1.1).

The Jacobi spectral collocation approach for (1.1) is to find $f_{\mathcal{L}}\in \mathcal{S}_{\mathcal{L}}^ \alpha$ , such that

$\begin{equation} \begin{split} \sum\limits_{i = 0}^{s} f^{(i)}_{\mathcal{L}}(x) = g_{\mathcal{L}}(x)+ \sum\limits_{i = 0}^{s}\mu_i(x) f^{(i)}_{\mathcal{L}}(c_ix)+\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}\nu_i(x) f^{(i)}_{\mathcal{L}}(w)dw+\int_{0}^{x}\xi(x) f_{\mathcal{L}}(w)dw. \end{split} \end{equation}$

(4.1)

Denote $f_{\mathcal{L}}(x)$ and $g_{\mathcal{L}}(x)$ by

$\begin{equation} f_{\mathcal{L}}(x) = \mathcal{F}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{equation}$

(4.2)

$\begin{equation} g_{\mathcal{L}}(x) = \mathcal{G}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{equation}$

(4.3)

with

$\mathcal{G}_\mathcal{L} = \left[g_0, g_1, \cdots, g_\mathcal{L}\right]^T; \qquad g_i = \frac{ \alpha^{a+b+1}}{2^{a+b+1}h_{ \alpha, i}^{a, b}}\sum\limits_{i = 0}^{\mathcal{L}} w^{(a, b)}_{\mathcal{L}, j} S_{ \alpha, i}^{(a, b)}\left(\frac{ \alpha}{2}\left(x^{(a, b)}_{\mathcal{L}, j}+1\right)\right) g\left(\frac{ \alpha}{2}\left(x^{(a, b)}_{\mathcal{L}, j}+1\right)\right).$

The following two theorems will be of great use later.

Theorem 3. ^[26] The $i^{th}$ derivative of the vector $\mathfrak{S}^{ \alpha}_{\mathcal{L}}(x)$ is given by:

$\begin{equation} \frac{d^i}{d x^i}\mathfrak{S}^{ \alpha}_{\mathcal{L}}(x) = \boldsymbol{D}^{(i)}\mathfrak{S}^{ \alpha}_{\mathcal{L}}(x); \qquad \boldsymbol{D}^{(i)} = (\boldsymbol{D}^{(1)})^i, \end{equation}$

(4.4)

with

$\boldsymbol{D}^{(1)} = (d_{i, j})_{0\leq i, j \leq \mathcal{L}}; \qquad d_{i, j} = \begin{cases}C_1(i, j), \ & j < i, \\ 0, &\mathit{\text{Otherwise}}, \end{cases}$

and

$\begin{split} C_1(i, j) = &\frac{ \alpha^{a+b}(i+a+b+1)(i+a+b+2)_j(j+a+2)_{i-j-1}\Gamma(j+a+b+1)}{(i-j-1)!\Gamma(2j+a+b+1)}\\ & \qquad \qquad \qquad \qquad\times{}_3F_2\left( \begin{array}{ccccc} -i+1+j, & i+j+a+b+2, & j+a+1 &\\ & & &;1\\ j+a+2, & 2j+a+b+2 & &\\ \end{array} \right). \end{split}$

Theorem 4. For $0 \leq c \leq 1,$ the pantograph operational matrix $\boldsymbol{P}_{c}$ is given by

$\begin{equation} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(c x) = \boldsymbol{P}_{c} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{equation}$

(4.5)

where

$\boldsymbol{P}_{c} = (\boldsymbol{p}^c_{l, j})_{0\leq l, j\leq \mathcal{L}};\qquad \boldsymbol{p}^c_{l, j} = \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^iq_{i, j}.$

Proof. We start by expressing $S_{ \alpha, l}^{a, b}(cx)$ by:

$\begin{equation} S_{ \alpha, l}^{(a, b)}(cx) = \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} \, c^ix^i. \end{equation}$

(4.6)

Expanding $x^i$ in terms of $S_{ \alpha, j}^{a, b}(x), \ j = 0, 1, \cdots, \mathcal{L},$ by

$\begin{equation} x^i = \sum\limits_{j = 0}^{\mathcal{L}}q_{i, j}S_{ \alpha, j}^{a, b}(x); \qquad \qquad q_{i, j} = \frac{1}{h_{ \alpha, j}^{(a, b)}} \int_{0}^{ \alpha} x^i S_{ \alpha, j}^{a, b}(x) w_ \alpha^{a, b}(x) dx. \end{equation}$

(4.7)

A combination of (4.6) and (4.7) then yields

$\begin{equation} \begin{split} S_{ \alpha, l}^{(a, b)}(cx)& = \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^i\left(\sum\limits_{j = 0}^{\mathcal{L}}q_{i, j}S_{ \alpha, j}^{a, b}(x)\right) = \sum\limits_{j = 0}^{\mathcal{L}} S_{ \alpha, j}^{a, b}\left(\sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^iq_{i, j}\right)\\ & = \left[\sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^iq_{i, 0}, \ \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^iq_{i, 1}, \ \cdots, \ \sum\limits_{i = 0}^{l} \mathcal{E}_{l, i} c^iq_{i, \mathcal{L}}\right]^T \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{split} \end{equation}$

(4.8)

which completes the proof. □

Application of (4.2) with Theorems 4.1 and 4.2, we have

$\begin{equation} \frac{d^i f_{\mathcal{L}}(x)}{dx^i} = \mathcal{F}^T_\mathcal{L} \frac{d^i }{dx^i} \left(\mathfrak{S}^{\alpha}_{\mathcal{L}}(x)\right) = \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \end{equation}$

(4.9)

$\begin{equation} \qquad\frac{d^i f_{\mathcal{L}}(cx)}{dx^i} = \mathcal{F}^T_\mathcal{L} \frac{d^i }{dx^i} \left(\mathfrak{S}^{\alpha}_{\mathcal{L}}(cx)\right) = \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)}\textbf{P}_{c} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x). \end{equation}$

(4.10)

Substitution from (4.2), (4.3), (4.9), and (4.10) into (4.1); one gets

$\begin{equation} \begin{split} \sum\limits_{i = 0}^{s} \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x) = & \sum\limits_{i = 0}^{s}\mu_i(x) \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)}\textbf{P}_{c_i} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x)+\int_{0}^{x}\xi(x)\mathcal{F}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw\\ &+\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}\nu_i(x) \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw+\mathcal{G}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x). \end{split} \end{equation}$

(4.11)

Thanks to (4.11), the residual $\mathcal{R}_{\mathcal{L}}(x)$ can be given by

$\begin{equation} \begin{split} \mathcal{R}_{\mathcal{L}}(x) = &\sum\limits_{i = 0}^{s} \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x)- \sum\limits_{i = 0}^{s}\mu_i(x) \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)}\textbf{P}_{c_i} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x)-\sum\limits_{i = 0}^{s}\int_{0}^{d_i x}\nu_i(x) \mathcal{F}^T_\mathcal{L} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw\\ &-\int_{0}^{x}\xi(x)\mathcal{F}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw-\mathcal{G}^T_\mathcal{L} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x). \end{split} \end{equation}$

(4.12)

Finally, the spectral solution of (1.1) is transformed to a problem of solving the following algebraic equations system:

$\begin{equation} \begin{split} &\mathcal{R}_{\mathcal{L}} (\frac{ \alpha}{2}(x_{\mathcal{L}, i}^{(a+b)}+1)) = 0, \qquad 0\leq i \leq \mathcal{L}-s, \\ &\mathcal{F}^T\textbf{D}^{(i)}\mathfrak{S}^{ \alpha}_{\mathcal{L}}(0) = \theta_i, \qquad i = 0, 1, \cdots, s-1. \end{split} \end{equation}$

(4.13)

5. System of multi-pantograph Volterra integro-differential equations

Here, the operational approach discussed in the previous section is carried out to get a numerical solution for the system of high-order multi-pantograph Volterra integro-differential equations:

$\begin{equation} \begin{split} \sum\limits_{k = 1}^{r} \gamma_{l, k}(x) f^{(s)}_k(x) = &g_l(x)+\sum\limits_{k = 1}^{r}\sum\limits_{i = 0}^{s}\mu_{l, k, i}(x) f^{(i)}_k (c_{i, l} x)+\sum\limits_{k = 1}^{r}\sum\limits_{i = 0}^{s}\int_{0}^{d_{i, l} x}\nu_{l, k, i}(x) f^{(i)}_k(w)dw\\ &+\int_{0}^{x}\xi_l(x) f_l(w)dw, \qquad x, w\in [0, \alpha], \quad 1\leq l \leq r, \end{split} \end{equation}$

(5.1)

with

$\begin{equation} f^{(i)}_l(0) = \theta_{l, i}, \qquad i = 0, 1, \cdots, s-1, \quad 1\leq l \leq r, \end{equation}$

(5.2)

where $r, \ s, \ 0\leq c_i, d_i\leq 1\ (0\leq i \leq s)$ are real numbers and $g_l(x),$ $\gamma_{k, i}(x),$ $\mu_{k, i}(x),$ $\nu_{k, i}(x),$ $\xi_{l}(x)$ $(0\leq k \leq r, \ 1\leq i \leq s)$ are known functions defined in $0\leq x\leq \alpha$ .

First, the system (5.1) may be written in the following way:

$\begin{equation} \begin{cases} &\textbf{A} \dfrac{d^s\textbf{F}(x)}{dx^s} = \textbf{G}(x) + \sum\limits_{i = 0}^{s} \textbf{B}_i \dfrac{d^i\textbf{F}(c_{i, l} x)}{dx^i}+ \sum\limits_{i = 0}^{s} \int_{0}^{d_{i, l}x} \textbf{C}_i \dfrac{d^i \textbf{F}(w)}{dw^i} dw+\int_{0}^{x}\textbf{E}\ \textbf{F}(w)dw, \\ & \dfrac{d^i\textbf{F}(0)}{dx^i} = \Theta_i, \qquad 0\leq i \leq s-1, \end{cases} \end{equation}$

(5.3)

where

$\begin{split} &\textbf{F}(x) = [f_1(x), f_2(x), \cdots, f_r(x)]^T, \\ &\textbf{G}(x) = [g_1(x), g_2(x), \cdots, g_r(x)]^T, \\ &\Theta_i = [\theta_{1, i}, \theta_{2, i}, \cdots, \theta_{r, i}]^T, \\ &\textbf{A} = (\gamma_{l, k})_{1\leq l \leq r, 1\leq k \leq r}, \qquad \textbf{B}_i = (\mu_{l, k, i})_{1\leq l \leq r, 1\leq k \leq r}, \qquad \textbf{C}_i = (\nu_{l, k, i})_{1\leq l \leq r, 1\leq k \leq r}, \\ &\textbf{E} = (e_{l, k})_{1\leq l \leq r, 1\leq k \leq r}, \qquad e_{l, l} = \xi_l, \quad e_{l, k} = 0\ \text{if}\ l\neq k. \end{split}$

Now, we aim to find the vector $\textbf{F}(x)$ such that:

$\begin{equation} \begin{split} &\textbf{A} \frac{d^s\textbf{F}_\mathcal{L}(x)}{dx^s} = \textbf{G}_\mathcal{L}(x) + \sum\limits_{i = 0}^{s} \textbf{B}_i \frac{d^i\textbf{F}_\mathcal{L}(c_{i, l} x)}{dx^i}+ \sum\limits_{i = 0}^{s} \int_{0}^{d_{i, l}x} \textbf{C}_i \frac{d^i \textbf{F}_\mathcal{L}(w)}{dw^i} dw+\int_{0}^{x}\textbf{E}\ \textbf{F}_\mathcal{L}(w)dw, \end{split} \end{equation}$

(5.4)

in this regard, we suppose

$\begin{equation} \begin{split} \textbf{F}_\mathcal{L}(x)& = \mathcal{F}_{\mathcal{L}, l} \mathfrak{S}^{\alpha}_{\mathcal{L}}(x); \qquad \mathcal{F}_{\mathcal{L}, l} = (\textbf{f}_{l, j})_{1\leq l\leq r, 0\leq j \leq \mathcal{L}}, \\ \textbf{G}_\mathcal{L}(x)& = \mathcal{G}_{\mathcal{L}, l} \mathfrak{S}^{\alpha}_{\mathcal{L}}(x); \qquad \ \mathcal{G}_{\mathcal{L}, l} = (\textbf{g}_{l, j})_{1\leq l\leq r, 0\leq j \leq \mathcal{L}}, \\ \end{split} \end{equation}$

(5.5)

where $\textbf{f}_{l, j}\ (1\leq l\leq r, 0\leq j \leq \mathcal{L})$ are the unknowns need to be determined later, while $\textbf{g}_{l, j}\ (1\leq l\leq r, 0\leq j \leq \mathcal{L})$ can be given using the shifted Jacobi Gauss quadrature rule as follows:

$\textbf{g}_{j, l} = \frac{ \alpha^{a+b+1}}{2^{a+b+1}h_{ \alpha, i}^{a, b}}\sum\limits_{i = 0}^{\mathcal{L}} w_{\mathcal{L}, i}^{(a+b)} S_{ \alpha, i}^{(a, b)}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right) g_l\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right), \qquad 1\leq l \leq r.$

Then one gets from (5.5) that

$\begin{equation} \frac{d^i \textbf{F}_\mathcal{L}(x)}{dx^i} = \mathcal{F}_{\mathcal{L}, l} \frac{d^i }{dx^i} \left(\mathfrak{S}^{\alpha}_{\mathcal{L}}(x)\right) = \mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \qquad \qquad 1\leq l \leq r, \quad 0\leq i \leq s, \end{equation}$

(5.6)

$\begin{equation} \frac{d^i \textbf{F}_\mathcal{L}(cx)}{dx^i} = \mathcal{F}_{\mathcal{L}, l} \frac{d^i }{dx^i} \left(\mathfrak{S}^{\alpha}_{\mathcal{L}}(cx)\right) = \mathcal{F}_{\mathcal{L}, l} \textbf{P}_{c} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x), \qquad 1\leq l \leq r, \quad 0\leq i \leq s. \end{equation}$

(5.7)

A combination of (5.5)–(5.7), Eq (5.4) can be approximated as:

$\begin{equation} \begin{split} \textbf{A} \mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(s)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x) = & \sum\limits_{i = 0}^{s} \textbf{B}_i \mathcal{F}_{\mathcal{L}, l} \textbf{P}_{c_{i, l}} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(x)+ \sum\limits_{i = 0}^{s} \int_{0}^{d_{i, l}x} \textbf{C}_i \mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w) dw\\ &+\int_{0}^{x}\textbf{E}\ \mathcal{F}_{\mathcal{L}, l} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw+\mathcal{G}_{\mathcal{L}, l} \mathfrak{S}^{\alpha}_{\mathcal{L}}(x). \end{split} \end{equation}$

(5.8)

Application of the spectral collocation method leads to the following system of $r (\mathcal{L}+1)$ algebraic equations:

$\begin{split} \textbf{A} \mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(s)} & \mathfrak{S}^{ \alpha}_{\mathcal{L}}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right) = \mathcal{G}_{\mathcal{L}, l} \mathfrak{S}^{\alpha}_{\mathcal{L}}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right)+ \sum\limits_{i = 0}^{s} \int_{0}^{d_{i, l}\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)} \textbf{C}_i \mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w) dw\\ &+\sum\limits_{i = 0}^{s} \textbf{B}_i \mathcal{F}_{\mathcal{L}, l} \textbf{P}_{c_{i, l}} \textbf{D}^{(i)} \mathfrak{S}^{ \alpha}_{\mathcal{L}}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right)+\int_{0}^{\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)}\textbf{E}\ \mathcal{F}_{\mathcal{L}, l} \mathfrak{S}^{ \alpha}_{\mathcal{L}}(w)dw, \qquad 1\leq l \leq r, \\ &\mathcal{F}_{\mathcal{L}, l} \textbf{D}^{(i)} \mathfrak{S}^{\alpha}_{\mathcal{L}}(0) = \Theta_{i}, \qquad 0\leq i \leq s-1, \quad 1\leq l \leq r. \end{split}$

6. Two-dimensional case

We utilize the Jacobi spectral collocation technique to solve the two-dimensional high-order multi-pantograph Volterra integro-differential equation (1.2).

As a spectral collocation scheme, we have to find $f_{\mathcal{L}, \mathcal{M}}\in \mathcal{S}_{\mathcal{L}}^ \alpha\times\mathcal{S}_{\mathcal{M}}^ \beta$ , such that

$\begin{equation} \begin{split} & \frac{\partial^{s+q} f_{\mathcal{L}, \mathcal{M}}(x, y) }{\partial x^{s}\partial y^{q}} = g_{\mathcal{L}, \mathcal{M}}(x, y)+ \sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\mu_{i, j}(x, y)\frac{\partial^{i+j} f_{\mathcal{L}, \mathcal{M}}(c_ix, d_jy)}{\partial x^{i}\partial y^{j}} \\ &+ \int_{0}^{y}\int_{0}^{x}\nu(t, u) f_{\mathcal{L}, \mathcal{M}}(t, u) dtdu + \sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\int_{0}^{c_i x}\int_{0}^{d_j y} \xi_{i, j}(t, u) \frac{\partial^{i+j} f_{\mathcal{L}, \mathcal{M}}(t, u)}{\partial t^{i}\partial u^{j}} dtdu . \end{split} \end{equation}$

(6.1)

In this regard, we suppose

$\begin{equation} \begin{split} f_{\mathcal{L}, \mathcal{M}}(x, y)& = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ g_{\mathcal{L}, \mathcal{M}}(x, y)& = \mathcal{G}^T_{\mathcal{L}, \mathcal{M}} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{split} \end{equation}$

(6.2)

where $\mathcal{F}_{\mathcal{L}, \mathcal{M}}$ is an unknown vector, while

$\mathcal{G}_{\mathcal{L}, \mathcal{M}} = [g_{l, m}, \qquad 0 \leq l \leq \mathcal{L}, \ 0 \leq m \leq \mathcal{M}]^T,$

$\begin{split} g_{l, m} = \frac{ \alpha^{a+b+1} \beta^{a+b+1}}{2^{2a+2b+2}h_{ \alpha, l}^{a, b}h_{ \beta, m}^{a, b}}\sum\limits_{l = 0}^{\mathcal{L}} \sum\limits_{m = 0}^{\mathcal{M}}& w_{\mathcal{L}, l}^{(a+b)} w_{\mathcal{M}, m}^{(a+b)} S_{ \alpha, l}^{(a, b)}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, l}^{(a+b)}+1\right)\right) S_{ \beta, m}^{(a, b)}\left(\frac{ \beta}{2}\left(x_{\mathcal{M}, m}^{(a+b)}+1\right)\right)\\ &\times g\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, l}^{(a+b)}+1\right), \frac{ \alpha}{2}\left(x_{\mathcal{M}, m}^{(a+b)}+1\right)\right). \end{split}$

Definition 1. ^[27] The kronecker product of any two $l \times m$ and $n \times o$ dimensional matrices $P$ and $Q,$ respectively, is denoted by the $ln\times mo$ matrix $P \otimes Q$ and is given by:

$P \otimes Q = \left( \begin{array}{cccc} p_{11} Q& p_{12} Q& \cdots & p_{1 m} Q\\ p_{21} Q& p_{22} Q& \cdots & p_{2 m} Q\\ \vdots & \vdots & \ddots & \vdots \\ p_{l 1} Q& p_{l 2} Q& \cdots & p_{lm} Q\\ \end{array} \right).$

Theorem 5. Assume $\mathcal{I}_{\mathcal{L}}$ and $\mathcal{I}_{\mathcal{M}}$ are the identity matrices of orders $\mathcal{L}+1$ and $\mathcal{M}+1,$ respectively; then

$\begin{equation} \begin{split} \frac{\partial^i}{\partial x^i} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y) & = \mathcal{D}^{(i)}_x\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{split} \end{equation}$

(6.3)

$\begin{equation} \begin{split} \frac{\partial^j}{\partial y^j} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y) & = \mathcal{D}^{(j)}_y\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{split} \end{equation}$

(6.4)

$\begin{equation} \begin{split} \frac{\partial^i}{\partial x^i} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(cx, y) & = \mathcal{D}^{(i)}_x \mathcal{P}_{x, c}\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{split} \end{equation}$

(6.5)

$\begin{equation} \begin{split} \frac{\partial^j}{\partial y^j} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, dy) & = \mathcal{D}^{(j)}_y\mathcal{P}_{y, d}\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \end{split} \end{equation}$

(6.6)

where $\mathcal{D}^{(i)}_x = \boldsymbol{D}^{(i)}\otimes \mathcal{I}_{\mathcal{L}}, \ \mathcal{D}^{(j)}_y = \mathcal{I}_{\mathcal{M}} \otimes \boldsymbol{D}^{(j)},$ $\mathcal{P}_{x, c} = \boldsymbol{P}_{c}\otimes \mathcal{I}_{\mathcal{M}},$ and $\mathcal{P}_{y, d} = \mathcal{I}_{\mathcal{L}}\otimes \boldsymbol{P}_{d}.$

In virtue of (6.2)–(6.6), we have

$\begin{equation} \begin{split} \frac{\partial^i}{\partial x^i} f_{\mathcal{L}, \mathcal{M}} (x, y) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(i)}_x\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ \frac{\partial^j}{\partial y^j} f_{\mathcal{L}, \mathcal{M}} (x, y) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(j)}_y\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ \frac{\partial^{i+j}}{\partial x^i \partial y^j} f_{\mathcal{L}, \mathcal{M}} (x, y) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(i)}_x \mathcal{D}^{(j)}_y\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ f_{\mathcal{L}, \mathcal{M}} (c x, y) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{P}_{x, c}\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ f_{\mathcal{L}, \mathcal{M}} (x, dy) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{P}_{y, d}\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y), \\ f_{\mathcal{L}, \mathcal{M}} (cx, dy) & = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{P}_{x, c}\mathcal{P}_{y, d}\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y). \\ \end{split} \end{equation}$

(6.7)

Then, the residual of (6.1) can be given by

$\begin{equation*} \begin{split} \mathcal{R}_{\mathcal{L}, \mathcal{M}} &(x, y) = \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(s)}_x \mathcal{D}^{(q)}_y\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y)-\mathcal{G}^T_{\mathcal{L}, \mathcal{M}} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y) \\ &- \sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\mu_{i, j}(x, y) \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(i)}_x \mathcal{D}^{(j)}_y\mathcal{P}_{x, c}\mathcal{P}_{y, d} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(x, y)-\int_{0}^{y}\int_{0}^{x}\nu(x, y) \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(t, u) dtdu \\ &-\sum\limits_{i = 0}^{s}\sum\limits_{j = 0}^{q}\int_{0}^{c_i x}\int_{0}^{d_j y} \xi_{i, j}(x, y) \mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(i)}_t \mathcal{D}^{(j)}_u\mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}(t, u) dtdu. \end{split} \end{equation*}$

Finally, we generate a system of $(\mathcal{L}+1)(\mathcal{M}+1)$ algebraic equations as follows:

$\begin{equation*} \begin{split} &\mathcal{R}_{\mathcal{L}, \mathcal{M}} \left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right), \frac{ \beta}{2}\left(x_{\mathcal{M}, j}^{(a+b)}+1\right)\right) = 0, \qquad \qquad \qquad \quad 0\leq i \leq \mathcal{L}-s, \ 0\leq j \leq \mathcal{M}-q, \\ &\mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(i)}_x \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}\left(0, \frac{ \beta}{2}\left(x_{\mathcal{M}, j}^{(a+b)}+1\right)\right) = \theta_i\left(\frac{ \beta}{2}\left(x_{\mathcal{M}, j}^{(a+b)}+1\right)\right), \qquad 0\leq i \leq s-1, \ 0\leq j \leq \mathcal{M}, \\ &\mathcal{F}^T_{\mathcal{L}, \mathcal{M}} \mathcal{D}^{(j)}_y \mathfrak{S}^{ \alpha, \beta}_{\mathcal{L}, \mathcal{M}}\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right), 0\right) = \vartheta_j\left(\frac{ \alpha}{2}\left(x_{\mathcal{L}, i}^{(a+b)}+1\right)\right), \qquad 0\leq j \leq q-1, \ 0\leq i \leq \mathcal{L}.\\ \end{split} \end{equation*}$

7. Test problems

The current section provides some test problems to ensure the validity of the numerical approaches presented in Sections 3–5. The programs used in this paper are performed using the PC machine, with CPU Intel(R) Core(TM) i3-2350M 2 Duo CPU 2.30 GHz and 6.00 GB of RAM. We also used the arithmetic symbolic program known as (Mathematica 12) to perform intermediate arithmetic operations and arithmetic tables, as well as illustrations in the paper as a whole.

7.1. Stability test

First, we consider the pantograph Volterra integro-differential equation ^[28,29]:

$\begin{equation} \begin{split} f^{(2)}(x)+&f\left(\frac{1}{2}x\right)-\frac{3}{4}f(x)-\int_{0}^{x}tf(t)dt = -\frac{11}{4}\sin(x)+x\cos(x)+\sin\left(\frac{x}{2}\right), \\ &f(0) = 0, \quad f'(0) = 1, \qquad 0\le x \leq 1, \end{split} \end{equation}$

(7.1)

with an exact solution $f(x) = \sin(x).$

To evaluate the stability of our proposed scheme, we introduce controlled perturbations into both the source term and the initial condition of Eq (7.1). This allows us to analyze how sensitive the method is to variations in input parameters. Consider the perturbed version of (7.1):

$\begin{equation} \begin{split} g^{(2)}(x) +& u\left(\frac{1}{2}x\right) - \frac{3}{4}g(x) - \int_{0}^{x} t g(t) dt = -\frac{11}{4}\sin(x) + x \cos(x) + \sin\left(\frac{x}{2}\right) + \epsilon_s, \\ &g(0) = 0, \qquad g'(0) = 1, \end{split} \end{equation}$

(7.2)

$\begin{equation} \begin{split} h^{(2)}(x) &+ u\left(\frac{1}{2}x\right) - \frac{3}{4}h(x) - \int_{0}^{x} t h(t) dt = -\frac{11}{4}\sin(x) + x \cos(x) + \sin\left(\frac{x}{2}\right), \\ &h(0) = \epsilon_i, \qquad h'(0) = 1+\epsilon_i, \end{split} \end{equation}$

(7.3)

where $\epsilon_s$ and $\epsilon_i$ are perturbations to the source term and the initial conditions, respectively. The exact solution remains $g(x) = h(x) = \sin(x)$ .

The numerical stability of the proposed scheme is analyzed by introducing perturbations to the source term ( $\epsilon_s$ ) in (7.2) and initial conditions ( $\epsilon_i$ ) in (7.3). Tables 1 and 2 show that both perturbations exhibit linear error propagation:

Table 1. MAEs for the perturbed Problem (7.2) with source term perturbations.

$\mathcal{L}$	$\epsilon_s=0.1$	$\epsilon_s=0.01$	$\epsilon_s=0.001$
6	0.05255861	0.00525315	0.00052261
8	0.05256080	0.00525608	0.00052561
10	0.05256079	0.00525607	0.00052560

| Show Table

DownLoad: CSV

Table 2. MAEs for the perturbed Problem (7.3) with initial condition perturbations.

$\mathcal{L}$	$\epsilon_i=0.1$	$\epsilon_i=0.01$	$\epsilon_i =0.001$
6	0.19715309	0.01971260	0.00196855
8	0.19715418	0.01971542	0.00197154
10	0.19715416	0.01971541	0.00197154

| Show Table

DownLoad: CSV

● For $\epsilon_s$ , reducing the perturbation magnitude by a factor of $10$ (e.g., $0.1 \to 0.01 \to 0.001$ ) decreases the maximum absolute errors (MAEs), $\lvert f_{\mathcal{L}}(x) - g_{\mathcal{L}}(x) \rvert$ , by the same factor.

● Similarly, for $\epsilon_i$ , the MAEs, $\lvert f_{\mathcal{L}}(x) - h_{\mathcal{L}}(x) \rvert$ , scale linearly with the perturbation size.

This linear behavior indicates the method's robustness against small perturbations in the source term or the initial condition.

Previous studies by Yüzbasi ^[28] (Laguerre operational matrix, LOM) and Gülsu & Sezer ^[29] (Taylor collocation method, TCM) provided approximate solutions. compares the numerical results of the absolute error function of $f(x)$ using the numerical method presented in Section 3 against those given using the LOM ^[28] and TCM ^[29]. plots the absolute errors of $f(x)$ with $\mathcal{L} = 10$ and the logarithmic function of MAEs for $f(x)$ at $(a, b) = (0, 0)$ .

Table 3. Absolute errors of

$f(x)$ with

$(a, b) = (1, 0)$ for Problem (7.1).

	LOM ^[28]		TCM ^[29]		Our scheme
$x$	$N=7$	$N=10$	$N=7$	$N=10$	$\mathcal{L}=7$	$\mathcal{L}=10$
0.0	$0.00$	$0.00$	$0.00$	$0.00$	$5.55\times10^{-17}$	$5.55\times10^{-17}$
0.2	$2.34\times10^{-9}$	$3.04\times10^{-13}$	$2.35\times10^{-7}$	$7.00\times10^{-10}$	$4.05\times10^{-10}$	$9.99\times10^{-15}$
0.4	$5.13\times10^{-9}$	$6.63\times10^{-13}$	$5.27\times10^{-7}$	$1.50\times10^{-9}$	$6.29\times10^{-10}$	$1.90\times10^{-14}$
0.6	$8.33\times10^{-9}$	$1.03\times10^{-12}$	$9.32\times10^{-7}$	$2.40\times10^{-9}$	$6.07\times10^{-10}$	$7.66\times10^{-15}$
0.8	$1.15\times10^{-8}$	$1.43\times10^{-12}$	$5.60\times10^{-7}$	$3.00\times10^{-9}$	$5.86\times10^{-10}$	$2.14\times10^{-14}$
1.0	$3.64\times10^{-9}$	$3.48\times10^{-12}$	$3.37\times10^{-5}$	$1.06\times10^{-7}$	$9.27\times10^{-12}$	$3.33\times10^{-16}$

| Show Table

DownLoad: CSV

Figure 1. The logarithmic function of MAEs and absolute errors of

$f(x)$ with

$(a, b) = (0, 0)$ and

$\mathcal{L} = 10$ for Problem (7.1).

DownLoad: Full-Size Img PowerPoint

7.2. Convergence test

To rigorously evaluate the convergence properties of the proposed method, we analyze the Volterra integro-differential equation:

$\begin{equation} f^{(1)}(x) - 3f(x) - f\left(\frac{1}{2}x\right) - \int_{0}^{x}2^x f(t)dt + 8\int_{0}^{\frac{1}{2}x}e^{x}\cos(t)f(t)dt = g(x), \quad 0 \leq x \leq 1, \end{equation}$

(7.4)

with $f(0) = 0$ and exact solution $f(x) = \sin(x)e^{2x-1}$ . This benchmark problem, previously studied in ^[15,17], allows direct comparison of convergence rates.

demonstrates the exponential convergence of our method compared to those achieved using the BMLW presented in ^[15], while compares the absolute errors of $f(x)$ with those given using the BT and LIT methods introduced in ^[17]. plots the absolute errors of $f(x)$ with $\mathcal{L} = 16$ and the logarithmic function of MAEs for $f(x)$ at $(a, b) = (1, 1)$ .

Table 4. MAEs of

$f(x)$ for Problem (7.4).

	$l=3$	$l=4$	$l=5$	$l=6$
BMLW ( $\mathcal{M}=3$ ) ^[15]	$6.563\times10^{-4}$	$8.508\times10^{-5}$	$1.080\times10^{-5}$	$1.359\times10^{-6}$
	$\mathcal{L}=6$	$\mathcal{L}=8$	$\mathcal{L}=10$	$\mathcal{L}=12$
Our method $(a, b)=(2, 2)$	$4.507\times10^{-4}$	$4.831\times10^{-6}$	$1.137\times10^{-8}$	$9.835\times10^{-12}$

| Show Table

DownLoad: CSV

Table 5. Absolute errors of

$f(x)$ at

$x = 0.5$ and 1 with

$(a, b) = (2, 0)$ for Problem (7.4).

	BT ^[17]		LIT ^[17]		Our method
$\mathcal{L}$	$x=0.5$	$x=1$	$x=0.5$	$x=1$	$x=0.5$	$x=1$
6	$2.75\times10^{-2}$	$3.17\times10^{-1}$	$1.85\times10^{-5}$	$4.20\times10^{-3}$	$2.47\times10^{-6}$	$6.79\times10^{-5}$
8	$1.99\times10^{-2}$	$2.46\times10^{-1}$	$2.00\times10^{-7}$	$2.30\times10^{4}$	$2.08\times10^{-8}$	$7.09\times10^{-7}$
10	$1.56\times10^{-2}$	$2.06\times10^{-1}$	$2.37\times10^{-9}$	$6.68\times10^{-6}$	$3.77\times10^{-11}$	$1.64\times10^{-9}$
12	$1.29\times10^{-2}$	$1.80\times10^{-1}$	$5.66\times10^{-11}$	$7.75\times10^{-7}$	$2.47\times10^{-14}$	$1.82\times10^{-12}$

| Show Table

DownLoad: CSV

Figure 2. The logarithmic function of MAEs and absolute errors of

$f(x)$ with

$(a, b) = (1, 1)$ and

$\mathcal{L} = 16$ for Problem (7.4).

DownLoad: Full-Size Img PowerPoint

7.3. Irregular solution test

To test the validity of the numerical approach presented in Section 3 for pantograph Volterra integro-differential equations with irregular solutions, we consider the following problem:

$\begin{equation} \begin{split} \frac{d^2 f(x)}{dx^2} + \frac{d f(x)}{dx} = \sin(x) f(x) - f\left(\frac{1}{3}x\right) + \int_{0}^{\frac{1}{2}x} t f(t) \, dt - \int_{0}^{x} \sqrt{t} f(t) \, dt + g(x), \quad 0 \leq x \leq 1, \end{split} \end{equation}$

(7.5)

where $f(0) = \frac{df}{dx}(0) = 0$ and $g(x)$ is chosen such that the exact solution is $f(x) = x^{\frac{7}{2}}$ .

To solve this problem, we apply the numerical scheme introduced in Section 3 with various choices of $\mathcal{L}$ . displays the MAEs of $f_{\mathcal{L}}(x)$ at $(a, b) = (0, 0)$ with $\mathcal{L} = 4, 8, 12, 16,$ and $20$ .

Table 6. MAEs of

$f_{\mathcal{L}}(x)$ for Problem (7.5).

$\mathcal{L}$	4	8	12	16	20
MAE	$1.1425\times10^{-3}$	$7.3101\times10^{-6}$	$1.0095\times10^{-6}$	$2.7518\times10^{-7}$	$9.8240\times10^{-8}$

| Show Table

DownLoad: CSV

7.4. System of equations

Consider the system of pantograph equations:

$\begin{equation} \begin{cases} \begin{aligned} f^{(1)}_1(x) & = \sin(x)f_1(x) - f_2(x) + 2f_1(cx) + \cos(x)f_2(cx) + \int_{0}^{x} f_1(t) \, dt - \int_{0}^{x} f_2(t) \, dt \\ &\quad + \int_{0}^{cx} x f_1(t) \, dt + \int_{0}^{cx} f_2(t) \, dt + g_1(x), \\[6pt] f^{(1)}_2(x) & = x f_1(x) - 4f_2(x) + f_1(cx) + e^{x}f_2(cx) + \int_{0}^{x} x^2 f_1(t) \, dt - \int_{0}^{x} \sin(x) f_2(t) \, dt \\ &\quad+ \int_{0}^{cx} 3f_1(t) \, dt + \int_{0}^{cx} f_2(t) \, dt + g_2(x), \\[6pt] &f_1(0) = 1, \quad f_2(0) = 0, \quad 0 \leq x \leq 1. \end{aligned} \end{cases} \end{equation}$

(7.6)

The exact solutions are given by $f_1(x) = e^{x}$ and $f_2(x) = x^3$ .

In ^[30], shifted Chebyshev polynomials were used as basis functions for the collocation spectral approach (CCS) to solve (7.6) with $c = 0.8$ . compares the MAEs of our method (using the numerical approach from Section 4 with $(a, b) = (0, 0)$ against the CCS results. shows the absolute error distributions for both components with $\mathcal{L} = 12$ , along with the logarithmic MAE plots for $(a, b) = (1, 0)$ with $c = 0.1$ .

Table 7. MAEs of

$f_1(x)$ and

$f_2(x)$ for Problem (7.6).

	CCS ^[30]		Our scheme
$\mathcal{L}$	$f_1(x)$	$f_2(x)$	$f_1(x)$	$f_2(x)$
2	$6.0777\times10^{-2}$	$1.1843\times10^{-1}$	$6.5075\times10^{-2}$	$4.8397\times10^{-2}$
4	$9.2227\times10^{-4}$	$8.2963\times10^{-4}$	$3.9735\times10^{-5}$	$6.6931\times10^{-5}$
6	$7.3973\times10^{-7}$	$1.8431\times10^{-6}$	$5.4490\times10^{-8}$	$2.8398\times10^{-9}$
8	$9.5045\times10^{-10}$	$6.6164\times10^{-10}$	$4.7068\times10^{-11}$	$1.9821\times10^{-12}$

| Show Table

DownLoad: CSV

Figure 3. The logarithmic functions of MAEs and absolute errors of

$f_1(x)$ and

$f_2(x)$ with

$(a, b) = (1, 0)$ and

$\mathcal{L} = 12$ for Problem (7.6).

DownLoad: Full-Size Img PowerPoint

7.5. Two-dimensional case

Consider the two-dimensional pantograph equation:

$\begin{equation} \begin{aligned} \frac{\partial^{3} f(x, y)}{\partial x^{2} \partial y} + f(x, y) &- f\left(x, \tfrac{1}{3}y\right) = \int_{0}^{y} \int_{0}^{x} f(s_1, s_2) \, ds_1 ds_2 + \int_{0}^{y} \int_{0}^{\frac{1}{2}x} f(s_1, s_2) \, ds_1 ds_2 \\ &\quad - \int_{0}^{\frac{1}{2}y} \int_{0}^{\frac{1}{2}x} \frac{\partial^{2} f(s_1, s_2)}{\partial x^{2}} \, ds_1 ds_2+ g(x, y) , \end{aligned} \end{equation}$

(7.7)

defined on the domain $0 \leq x, y \leq 1$ , with exact solution $f(x, y) = (x+1)\sin(y)$ .

To validate the numerical scheme from Section 5, presents the absolute errors of $f(x, y)$ for different values of $\mathcal{L}$ and $\mathcal{M}$ with $(a, b) = (2, 2)$ . shows the error distribution for $\mathcal{L} = \mathcal{M} = 10$ at $(a, b) = (3, 3)$ , while displays corresponding contour plots for various values of $\mathcal{L}$ and $\mathcal{M}$ .

Table 8. Absolute errors of

$f(x, y)$ with

$(a, b) = (2, 2)$ for Problem (7.7).

$(x_1, x_2)$	$(\mathcal{L}, \mathcal{M})=(2, 2)$	$(\mathcal{L}, \mathcal{M})=(4, 4)$	$(\mathcal{L}, \mathcal{M})=(6, 6)$	$(\mathcal{L}, \mathcal{M})=(8, 8)$	$(\mathcal{L}, \mathcal{M})=(10, 10)$
(0.1, 0.1)	$4.7780\times10^{-3 }$	$7.5011\times10^{-6 }$	$8.7689\times10^{-8 }$	$1.8082\times10^{-10}$	$1.8707\times10^{-13}$
(0.2, 0.2)	$1.0733\times10^{-3 }$	$4.6338\times10^{-5 }$	$1.7930\times10^{-7 }$	$3.0610\times10^{-10}$	$2.7519\times10^{-13}$
(0.3, 0.3)	$4.8745\times10^{-3 }$	$6.9301\times10^{-5 }$	$2.4872\times10^{-7 }$	$3.6300\times10^{-10}$	$2.8521\times10^{-13}$
(0.4, 0.4)	$7.1270\times10^{-3 }$	$9.2293\times10^{-5 }$	$2.8985\times10^{-7 }$	$3.7009\times10^{-10}$	$3.0797\times10^{-13}$
(0.5, 0.5)	$8.6790\times10^{-3 }$	$1.1553\times10^{-4 }$	$2.9407\times10^{-7 }$	$3.9560\times10^{-10}$	$3.2218\times10^{-13}$
(0.6, 0.6)	$1.0695\times10^{-2 }$	$1.2945\times10^{-4 }$	$2.9661\times10^{-7 }$	$4.1656\times10^{-10}$	$3.3228\times10^{-13}$
(0.7, 0.7)	$1.4623\times10^{-2 }$	$1.2567\times10^{-4 }$	$3.2109\times10^{-7 }$	$4.1231\times10^{-10}$	$3.4572\times10^{-13}$
(0.8, 0.8)	$2.2154\times10^{-2 }$	$1.1240\times10^{-4 }$	$3.2223\times10^{-7 }$	$4.3780\times10^{-10}$	$3.4061\times10^{-13}$
(0.9, 0.9)	$3.5178\times10^{-2 }$	$1.3403\times10^{-4 }$	$2.8893\times10^{-7 }$	$4.0347\times10^{-10}$	$3.5038\times10^{-13}$

| Show Table

DownLoad: CSV

Figure 4. Absolute error function of

$f(x, y)$ with

$(a, b) = (3, 3)$ and

$\mathcal{L} = \mathcal{M} = 10$ for Problem (7.7).

DownLoad: Full-Size Img PowerPoint

Figure 5. The contour plot of absolute errors of

$f(x, y)$ with

$(a, b) = (3, 3)$ for Problem (7.7).

DownLoad: Full-Size Img PowerPoint

8. Conclusions

This paper dealt with a problem of great importance in physics and engineering, namely, high-order multi-pantograph Volterra integro-differential equations with variable coefficients. The pantograph and integral terms enable the equation to study complex systems with memory effects and scaling. We investigated the existence and uniqueness, for the first time, of the solution to the high-order multi-pantograph Volterra integro-differential equation. We introduced the pantograph operational matrix, for the first time based on shifted Jacobi polynomials, and employed it with the operational matrix of differentiation to convert the studied problem into a system of algebraic equations with the aid of the spectral collocation method. The studied numerical approach is also applied for the system of high-order multi-pantograph Volterra integro-differential equations and for high-order two-dimensional multi-pantograph Volterra integro-differential equations with variable coefficients. Comparing the numerical results of five test problems with the exact solution and with other spectral methods applied to the same problems confirms the superiority of the new one. The studied numerical approach is shown to be more accurate than the Laguerre operational matrix, Taylor collocation, Chebyshev collocation spectral, Bernoulli müntz–Legendre wavelet, Lagrange interpolation, and Bernstein tau methods. At the end, we note that the presented approach can be extended to more complex integro-differential equations. This will be the subject of our future research.

Author contributions

A.H. Tedjani: Validation, Methodology, Writing the original draft; S.E. Alhazmi: Validation, Methodology, Writing the original draft; S.S. Ezz-Eldien: Writing–review & editing, Writing–original draft, Validation, Supervision, Software, Investigation. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

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

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).

Conflicts of interest

The authors declare that they have no conflicts of interest.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).

References

[1]	S. Ghosh, J. Heffernan, Influenza Pandemic Waves under Various Mitigation Strategies with 2009 H1N1 as a Case Study, PLoS One, 5 (2010), e14307. http://dx.doi.org/10.1371/journal.pone.0014307 doi: 10.1371/journal.pone.0014307
[2]	A. R. Tuite, D. N. Fisman, A. L. Greer, Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada, Cmaj, 192 (2020), E497–E505.
[3]	B. M. Althouse, S. V. Scarpino, L. A. Meyers, J. W. Ayers, M. Bargsten, J. Baumbach, et al., Enhancing disease surveillance with novel data streams: challenges and opportunities, EPJ Data Sci., 4 (2015), 1–8. http://dx.doi.org/10.1140/epjds/s13688-015-0054-0 doi: 10.1140/epjds/s13688-015-0054-0
[4]	M. Roser, H. Ritchie, E. Ortiz-Ospina, J. Hasell, Coronavirus Pandemic (COVID-19), Our World in Data, 2020. Available from: https://ourworldindata.org/covid-cases
[5]	T. Oraby, O. Vasilyeva, D. Krewski, F. Lutscher, Modeling seasonal behavior changes and disease transmission with application to chronic wasting disease, J. Theor. Biol., 340 (2014), 50–59. http://dx.doi.org/10.1016/j.jtbi.2013.09.003 doi: 10.1016/j.jtbi.2013.09.003
[6]	A. Mummert, H. Weiss, L.-P. Long, J. M. Amigo, X.-F. Wan, A Perspective on Multiple Waves of Influenza Pandemics, PLoS One, 8 (2013), e60343. http://dx.doi.org/10.1371/journal.pone.0060343 doi: 10.1371/journal.pone.0060343
[7]	N. C. Grassly, C. Fraser, Seasonal infectious disease epidemiology, P. Roy. Soc. B - Biol. Sci., 273 (2006), 2541–2550. http://dx.doi.org/10.1098/rspb.2006.3604 doi: 10.1098/rspb.2006.3604
[8]	W. P. London, J. A. Yorke, Recurrent outbreaks of measles, chickenpox, and mumps.1. Seasonal-variation in contact rates, Am. J. Epidemiol., 98 (1973), 453–468. http://dx.doi.org/10.1093/oxfordjournals.aje.a121575 doi: 10.1093/oxfordjournals.aje.a121575
[9]	D. Glabska, D. Skolmowska, D. Guzek, Population-based study of the influence of the COVID-19 pandemic on hand hygiene behaviors - Polish adolescents' COVID-19 experience (PLACE-19) study, Sustainability, 12 (2020), 4930. https://doi.org/10.3390/su12124930 doi: 10.3390/su12124930
[10]	C. O'Connor, J. O. Weatherall, The misinformation age: How false beliefs spread, Yale University Press, 2019. https://doi.org/10.2307/j.ctv8jp0hk
[11]	E. Dubois, G. Blank, The echo chamber is overstated: the moderating effect of political interest and diverse media, Inf., commun. soc., 21 (2018), 729–745. https://doi.org/10.1080/1369118X.2018.1428656 doi: 10.1080/1369118X.2018.1428656
[12]	D. M. Kahan, A risky science communication environment for vaccines, Science, 342 (2013), 53–54. https://doi.org/10.1126/science.1245724 doi: 10.1126/science.1245724
[13]	B. Baumgaertner, B. J. Ridenhour, F. Justwan, J. E. Carlisle, C. R. Miller, Risk of disease and willingness to vaccinate in the United States: A population-based survey, PLoS med., 17 (2020), e1003354. https://doi.org/10.1371/journal.pmed.1003354 doi: 10.1371/journal.pmed.1003354
[14]	I. C.-H. Fung, S. Cairncross, How often do you wash your hands? A review of studies of hand-washing practices in the community during and after the SARS outbreak in 2003, Int. J. Environ. Heal. R., 17 (2007), 161–183. https://doi.org/10.1080/09603120701254276 doi: 10.1080/09603120701254276
[15]	S. Jamshidi, M. Baniasad, D. Niyogi, Global to USA country scale analysis of weather, urban density, mobility, homestay, and mask use on COVID-19, Int. J. Env. Res. Pub. Heal., 17 (2020), 7847. http://dx.doi.org/10.3390/ijerph17217847 doi: 10.3390/ijerph17217847
[16]	A. Matakos, E. Terzi, P. Tsaparas, Measuring and moderating opinion polarization in social networks, Data Min. Knowl. Disc., 31 (2017), 1480–1505. https://doi.org/10.1007/s10618-017-0527-9 doi: 10.1007/s10618-017-0527-9
[17]	J.-M. Esteban, D. Ray, On the mesaurement of polarization, econometrica, 62 (1994), 819–851. https://doi.org/10.2307/2951734 doi: 10.2307/2951734
[18]	P. Manfredi, A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer, New York, 2017. http://dx.doi.org/10.1007/978-1-4614-5474-8
[19]	N. Perra, D. Balcan, B. Gonçalves, A. Vespignani, Towards a characterization of behavior-disease models, PloS one, 6 (2011), e23084. https://doi.org/10.1371/journal.pone.0023084 doi: 10.1371/journal.pone.0023084
[20]	L. Mao, Predicting self-initiated preventive behavior against epidemics with an agent-based relative agreement model, J. Artif. Soc. Social S., 18 (2015), 6. https://doi.org/10.18564/jasss.2892 doi: 10.18564/jasss.2892
[21]	A. Mummert, H. Weiss, Get the news out loudly and quickly: the influence of the media on limiting emerging infectious disease outbreaks, PloS one, 8 (2013), e71692. https://doi.org/10.1371/journal.pone.0071692 doi: 10.1371/journal.pone.0071692
[22]	P. Poletti, B. Caprile, M. Ajelli, A. Pugliese, S. Merler, Spontaneous behavioural changes in response to epidemics, J. Theor. Biol., 260 (2009), 31–40. https://doi.org/10.1016/j.jtbi.2009.04.029 doi: 10.1016/j.jtbi.2009.04.029
[23]	P. Poletti, M. Ajelli, S. Merler, Risk perception and effectiveness of uncoordinated behavioral responses in an emerging epidemic, Math. Biosci., 238 (2012), 80–89. https://doi.org/10.1016/j.mbs.2012.04.003 doi: 10.1016/j.mbs.2012.04.003
[24]	S. Maharaj, A. Kleczkowski, Controlling epidemic spread by social distancing: Do it well or not at all, BMC Public Health, 12 (2012), 1–16. https://doi.org/10.1186/1471-2458-12-679 doi: 10.1186/1471-2458-12-679
[25]	M. Martcheva, An Introduction to Mathematical Epidemiology, Vol. 61, Springer, 2015. https://doi.org/10.1007/978-1-4899-7612-3
[26]	S. C. Anderson, A. M. Edwards, M. Yerlanov, N. Mulberry, J. E. Stockdale, S. A. Iyaniwura, et al., Estimating the impact of COVID-19 control measures using a Bayesian model of physical distancing, medRxiv, 2020. https://doi.org/10.1101/2020.04.17.20070086 doi: 10.1101/2020.04.17.20070086
[27]	N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, M. Baguelin, et al., Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College London, (2020). https://doi.org/10.25561/77482 doi: 10.25561/77482
[28]	L. G. Nardin, C. R. Miller, B. J. Ridenhour, S. M. Krone, P. Joyce, B. O. Baumgaertner, Planning horizon affects prophylactic decision-making and epidemic dynamics, PeerJ, 4 (2016), e2678. https://doi.org/10.7717/peerj.2678 doi: 10.7717/peerj.2678
[29]	D. Weston, K. Hauck, R. Amlôt, Infection prevention behaviour and infectious disease modelling: a review of the literature and recommendations for the future, BMC Public Health, 18 (2018), 336–351. https://doi.org/10.1186/s12889-018-5223-1 doi: 10.1186/s12889-018-5223-1
[30]	R. C. Tyson, S. D. Hamilton, A. S. Lo, B. O. Baumgaertner, S. M. Krone, The Timing and Nature of Behavioural Responses Affect the Course of an Epidemic, B. Math. Biol., 82 (2020), 1–28. https://doi.org/10.1007/s11538-019-00684-z doi: 10.1007/s11538-019-00684-z
[31]	A. K. Misra, R. K. Rai, Y. Takeuchi, Modeling the control of infectious diseases: Effects of TV and social media advertisements, Math. Biosci. Eng., 15 (2018), 1315–1343. https://doi.org/10.3934/mbe.2018061 doi: 10.3934/mbe.2018061
[32]	MATLAB, 9.8.0.1323502 (R2020a), The MathWorks Inc., Natick, Massachusetts, 2020.
[33]	M. J. Keeling, L. Danon, Mathematical modelling of infectious diseases, Brit. Med. Bull., 92 (2009), 33–42. https://doi.org/10.1093/bmb/ldp038 doi: 10.1093/bmb/ldp038
[34]	A. Huppert, G. Katriel, Mathematical modelling and prediction in infectious disease epidemiology, Clin. Microbiol. Infec., 19 (2013), 999–1005. https://doi.org/10.1111/1469-0691.12308 doi: 10.1111/1469-0691.12308
[35]	M. W. Fong, H. Gao, J. Y. Wong, J. Xiao, E. Y. C. Shiu, S. Ryu, et al., Nonpharmaceutical Measures for Pandemic Influenza in Nonhealthcare Settings—Social Distancing Measures, Emerg. Infect. Dis., 26 (2020), 976–984. https://doi.org/10.3201/eid2605.190995 doi: 10.3201/eid2605.190995
[36]	A. E. Aiello, R. M. Coulborn, V. Perez, E. L. Larson, Effect of Hand Hygiene on Infectious Disease Risk in the Community Setting: A Meta-Analysis, Am. J. Public Health, 98 (2008), 1372–1381. https://doi.org/10.2105/ajph.2007.124610 doi: 10.2105/ajph.2007.124610
[37]	T. Jefferson, R. Foxlee, C. Del Mar, L. Dooley, E. Ferroni, B. Hewak, et al., Physical interventions to interrupt or reduce the spread of respiratory viruses: systematic review, BMJ, 336 (2007), 77–80. https://doi.org/10.1136/bmj.39393.510347.be doi: 10.1136/bmj.39393.510347.be
[38]	F. Chen, A. Griffith, A. Cottrell, Y.-L. Wong, Behavioral Responses to Epidemics in an Online Experiment: Using Virtual Diseases to Study Human Behavior, PLoS ONE, 8 (2013), e52814. https://doi.org/10.1371/journal.pone.0052814
[39]	YouGov, Personal measures taken to avoid COVID-19, 2020. Available from: https://today.yougov.com/topics/international/articles-reports/2020/03/17/personal-measures-taken-avoid-covid-19
[40]	AP-NORC, Coronavirus Concerns on the Rise Again Amid Spike in New Cases, 2020. Available from: https://apnorc.org/projects/coronavirus-concerns-on-the-rise-again-amid-spike-in-new-cases/?doing_wp_cron=1598305841.2085039615631103515625

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)