On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

Ridha Dida; Hamid Boulares; Bahaaeldin Abdalla; Manar A. Alqudah; Thabet Abdeljawad; Ridha Dida; Hamid Boulares; Bahaaeldin Abdalla; Manar A. Alqudah; Thabet Abdeljawad

doi:10.3934/math.20231172

AIMS Mathematics

2023, Volume 8, Issue 10: 23032-23045. doi: 10.3934/math.20231172

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Research article Special Issues

On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

1.
Depatement of Mathematics, Faculty of Sciences, University Badji Mokhtar Annaba, P.O. Box 12, Annaba, 23000, Algeria
2.
Laboratory of Analysis and Control of Differential Equations "ACED'', University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea

Received: 24 May 2023 Revised: 02 July 2023 Accepted: 06 July 2023 Published: 19 July 2023
MSC : 26A33, 34B10, 34B15

Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $\Psi$ (shortly $\Psi$ -Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.

Keywords:

Citation: Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad. On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives[J]. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172

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Abstract

1. Introduction

The study on flow of fluids which are electrically conducting is known as magnetohydrodynamics (MHD). The magnetohydrodynamics have important applications in the polymer industry and engineering fields (Garnier ^[1]). Heat transfer caused by hydromagnetism was discussed by Chakrabarthi and Gupta ^[2]. Using an exponentially shrinking sheet, Nadeem et al. ^[3] investigated the MHD flow of a Casson fluid. Krishnendu Bhattacharyya ^[4] examined the effect of thermal radiation on MHD stagnation-point Flow over a Stretching Sheet.

Mixed convection magnetohydrodynamics flow is described by Ishak on a vertical and on a linearly stretching sheet ^[5,6,7]. Hayat et al. ^[10] examine a mixed convection flow within a stretched sheet of Casson nanofluid. Subhas Abel and Monayya Mareppa, examine magnetohydrodynamics flow on a vertical plate ^[11]. Shen et al. ^[12], examined a vertical stretching sheet which was non-linear. Ishikin Abu Bakar ^[13] investigates how boundary layer flow is affected by slip and convective boundary conditions over a stretching sheet. A vertical plate oscillates with the influence of slip on a free convection flow of a Casson fluid ^[14].

Nasir Uddin et al. ^[15] used a Runge-Kutta sixth-order integration method. Barik et al. ^[16] implicit finite distinction methodology of Crank Sir Harold George Nicolson sort Raman and Kumar ^[17] utilized an exact finite distinction theme of DuFort–Frankel. Mondal et al. ^[18] used a numerical theme over the whole vary of physical parameters. With the laplace transform method, we can determine the magnetohydrodynamic flow of a viscous fluid ^[19]. Thamizh Suganya et al. ^[20] obtained that the MHD for the free convective flow of fluid is based on coupled non-linear differential equations. In this study, the analytical approximation of concentration profiles in velocity, temperature and concentration using homotopy perturbation method (HPM).

2. Mathematical construction of the problem

The governing differential equations in dimensionless form ^[19] as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0, \end{align}$

(2.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0, \end{align}$

(2.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0. \end{align}$

(2.3)

The dimensionless boundary conditions given by:

$\begin{equation} u = 0, \ \theta = 1, \phi = 0 \ \ at \ \ y = 0 \end{equation}$

(2.4)

and

$\begin{equation} u = 0, \ \theta = 0, \phi = 0 \ \ at \ \ y \to \infty. \end{equation}$

(2.5)

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

He ^[21,22] established the homotopy perturbation method, which waives the requirement of small parameters. Many researchers have used HPM to obtain approximate analytical solutions for many non-linear engineering dynamical systems ^[23,24]. The basic concept of the HPM as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0 \end{align}$

(3.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0 \end{align}$

(3.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0 \end{align}$

(3.3)

with initial and boundary conditions given by:

$\begin{align} y = 0 \ \ &at \ u = 0, \ \theta = 1, C = 1 \\ y \to \infty \ \ &as \ u = 0, \ \theta = 1, \phi = 0. \end{align}$

(3.4)

Homotopy for the above Eqs (3.1) to (3.4) can be constructed as follows:

$\begin{align} (1 - p) \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] + p \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] = 0 \end{align}$

(3.5)

$\begin{align} (1 - p) \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta\right] + p \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta + \theta\right] = 0 \end{align}$

(3.6)

$\begin{align} (1 - p) \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - \gamma \phi \right] + p \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \theta}{\partial y^2} - \gamma \phi \right] = 0 \end{align}$

(3.7)

The approximate solution of the Eqs (3.5) to (3.7) are

$\begin{align} u & = u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ... \end{align}$

(3.8)

$\begin{align} \theta & = \theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ... \end{align}$

(3.9)

$\begin{align} \phi & = \phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ... \end{align}$

(3.10)

Substitution Eqs (3.5) to (3.7) in Eqs (3.8) to (3.10) respectively. We obtain the following equations

$\begin{align} (1 - p) &\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...) \right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)\right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] = 0 \end{align}$

(3.11)

and

$\begin{align} (1 - p) &\left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] \\ &+ p \left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) \right. \\ &\left.+ (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] = 0, \end{align}$

(3.12)

and

$\begin{align} (1 - p) &\left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p \left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - Sr \frac{\partial^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{\partial y^2} \right. \\ &\left.- \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...) \right] = 0. \end{align}$

(3.13)

Equating the coefficient of $p$ on both sides, we get the following equations

$\begin{align} P^0 &: \frac{d^2 u_0}{dy^2} - Hu_0 + Gr \theta_0 + Gm \phi_0 = 0; \end{align}$

(3.14)

$\begin{align} p^0 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 = 0; \end{align}$

(3.15)

$\begin{align} P^1 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 + \theta_1 = 0; \end{align}$

(3.16)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - \gamma \phi_0 = 0; \end{align}$

(3.17)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - Sr \frac{\partial^2 \theta_0}{\partial y^2} - \gamma \phi_0 = 0. \end{align}$

(3.18)

The boundary conditions are

$\begin{align} u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y = 0 \\ u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y \to \infty. \end{align}$

(3.19)

and

$\begin{align} u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y = 0 \\ u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y \to \infty. \end{align}$

(3.20)

Solving the Eqs (3.9)–(3.14), we obtain

$\begin{align} u_0(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.21)

$\begin{align} \theta_0(y) & = e^{-y \sqrt{\frac{1}{F}}}; \end{align}$

(3.22)

$\begin{align} \phi_0(y) & = e^{-y \sqrt{\gamma Sc}}; \end{align}$

(3.23)

$\begin{align} \phi_1(y) & = \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.24)

Considering the iteration, we get,

$\begin{align} u(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.25)

$\begin{align} \theta(y) & = e^{-y \sqrt{F}}; \end{align}$

(3.26)

$\begin{align} \phi(y) & = e^{-y \sqrt{\gamma Sc}} + \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.27)

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

From the Eqs (3.25)–(3.27), we obtain

$\begin{array}{l} C_f = - \left(\frac{\partial u}{\partial y}\right)_{y = 0} \\ = \frac{\left(Gm \sqrt{\gamma Sc} - Gm \sqrt{H} + Gr(\gamma Sc + H) \sqrt{1 + \frac{R}{Pr}} + Gm H \sqrt{\gamma Sc} + (-Gm - Gr) H^{\frac{3}{2}} - \gamma Gr \sqrt{H} Sc\right)}{\left(\sqrt{1 + \frac{R}{Pr} + H} (\gamma Sc + H)\right)}; \end{array}$

(4.1)

$\begin{align} N_u & = - \left(\frac{\partial \theta}{\partial y}\right)_{y = 0} = \sqrt{\frac{1 + R}{Pr}}; \end{align}$

(4.2)

$\begin{array}{l} S_h = - \left(\frac{\partial \phi}{\partial y}\right)_{y = 0} \\= \frac{Sc Sr (1 + R) \sqrt{1 + \frac{R}{Pr}} - \sqrt{\gamma Sc} \left((-R - 1) \sqrt{1 + \frac{R}{Pr}} + ((1 + R)Sc - Pr \gamma) Sc \right)}{(1 + R) \sqrt{1 + \frac{R}{Pr}} Pr \gamma Sc} \end{array}$

(4.3)

5. Results and discussion

The combined impacts of transient MHD free convective flows of an incompressible viscous fluid through a vertical plate moving with uniform motion and immersed in a porous media are examined using an exact approach. The approximate analytical expressions for the velocity $u$ , temperature $\theta$ , and concentration profile $\phi$ are solved by using the homotopy perturbation method for fixed values of parameters is graphically presented.

Velocity takes time at first, and for high values of y, it takes longer, and the velocity approaches zero as time increases. The velocity of fluid rises with Gr increasing, as exposed in Figure 1.

Figure 1. An illustration of velocity profiles for different values of Gr.

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AIMS Mathematics

On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

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Abstract

1. Introduction

2. Mathematical construction of the problem

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

5. Results and discussion

6. Conclusions

Conflict of interest

Acknowledgement

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