On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

Iman Ben Othmane; Lamine Nisse; Thabet Abdeljawad; Iman Ben Othmane; Lamine Nisse; Thabet Abdeljawad

doi:10.3934/math.2024686

AIMS Mathematics

2024, Volume 9, Issue 6: 14106-14129. doi: 10.3934/math.2024686

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Research article

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

1.
Department of Mathematics, Faculty of Exact Sciences, Operators Theory and PDE Foundations and Applications Laboratory, University of El-Oued, P.O. Box 789, El-Oued 39000, Algeria
2.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea
5.
Department of Mathematics and Applied Mathematics School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 18 January 2024 Revised: 23 March 2024 Accepted: 02 April 2024 Published: 18 April 2024
MSC : 26A33, 34A08, 34K37

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $\delta$ , $0 < \delta < 1$ .

Keywords:

Citation: Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad. On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems[J]. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686

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Abstract

1. Introduction

In the literature of fractional calculus, we have lots of operators of fractional derivative as: the Hillfer-Prabhakar derivatives ^[1], the Caputo-Fabrizio fractional derivative ^[2,3]. In recent years fractional calculus has received considerable attention to solving the mathematics, engineering, mathematical physics and biological problems. Fractional calculus has been playing a Vitol roll in the viscoelastic and diffusion models. In ^[4], Hristov works on integral solution of fractional subdivisional differential equation by applying integral approach. In ^[5,6], Hristov established the relation between Caputo-Fabrizio fractional derivative and Cattaneo heat diffusion equation with Jeffrey's kernel the derivative of heat diffusion equation. Hristov was the first researcher who developed the physical importance of the recently defined Caputo-Fabrizio derivative with non-singular kernel in heat diffusion equation. In ^[7], Hristov used recently developed Atangana-Baleanu fractional derivative and Mittag-leffler function to construct adequate physical form of heat diffusion equation, for others models see in ^[8].

In the classical diffusion equation we obtained diffusion equation of fractional order when just replacing ordinary derivative by a specific fractional derivative operator. In this work, we use the Hilfer-Prabhakar derivative. Here, we present the solution in analytical form of one and two-dimensional spaces of the fractional diffusion equations. Koca et al. ^[9] generalized the Hristov model of elastic heat diffusion (EHD) equation and solved it explicitly difference scheme and also numerical approximation for first and second order approximation was introduced. Alkahtani et al. ^[10] introduced solution in terms of numerical results of the CCHD (complete Cattaneo-Hristov diffusion) equation by applying the numerical scheme "Crank-Nicholson". Hristov ^[11] introduced an approximate solution of the Cattaneo-Hristov diffusion (CHD) equation by using many more method like the "heat-balance integral method" (HBIM) and "Double integral-balance method" and also to obtained the approximate solution of the CHD equation using HBIM and DIM were proposed in ^{[11,12,13,14,15,16,17,18]}. In ^[9] Koca et al gives the analytical solution of the CHD equation. In this article, we continue the work concerning the analytical solution stated by Koca et al. in ^[9]. In this article, we investigate complete Cattaneo-Hristov diffusion (CCHD) equation and FDE (fractional diffusion equation) in one and two dimensional spaces and find their analytic solution under the Dirichlet boundary conditions by using an integral transform method. This method uses both the Fourier sine transform and the Elzaki transform. With the help of this method we demonstrate the analytical solutions of the fractional diffusion equations (FDE) in term of the Mittag-Leffer function ^[19,20].

2. Definitions

In this portion, we study important definitions related to fractional calculus and Elzaki transform to understand the further results.

The Elzaki transform ^[21] of the function f(t) is given by

$E\left[ {f\left( t \right)} \right] = T\left( v \right) = v\int_0^\infty {} f\left( t \right){e^{ - \frac{t}{v}}}d{\rm{t}}, t \gt 0, v \in \left( { - {k_1}, {k_2}} \right)$

(2.1)

then the Elzaki transform of the convolution of f(t) and g(t) is express as

$E\left[ {\left( {f*g} \right)\left( t \right)} \right] = \frac{1}{v}M\left( v \right)N\left( v \right)$

(2.2)

Where

$\left( {f*g} \right)\left( t \right) = \int_0^t {} f\left( {x - t} \right)g\left( t \right)dt$

(2.3)

Whenever the integral is defined.

Caputo-Fabrizio ^[2] fractional time derivative is defined as

$D_t^\alpha f\left( t \right) = \frac{{M\left( \alpha \right)}}{{\left( {1 - \alpha } \right)}}\int_a^t {} \dot f\left( t \right){e^{\left[ {\frac{{ - \alpha \left( {t - \tau } \right)}}{{1 - \alpha }}} \right]}}d\tau$

(2.4)

We suppose the function ${M\left(\alpha \right)}$ and a = 0

Elzaki transform of the Caputo-Fabrizio fractional derivative

$E\left[ {D_t^\alpha f\left( t \right)} \right] = \frac{1}{{\left( {1 - \alpha } \right)}}v\int_0^\infty {} {e^{ - \frac{t}{v}}}\mathop \smallint \limits_a^t \dot f\left( t \right){e^{\left[ {\frac{{ - \alpha \left( {t - \tau } \right)}}{{1 - \alpha }}} \right]}}d\tau dt$

The convolution property of Elzaki transform is defined as

$\begin{array}{l} E\left[D_{t}^{\alpha} f(t)\right] = \frac{v}{(1-\alpha)} \cdot E[\dot{f}(t)] *\left(e^{\left.\frac{-\alpha t}{(1-\alpha)}\right)}\right.\\ = \frac{v}{(1-\alpha)} \cdot \frac{1}{v}\left[\frac{T(v)}{v}-v(f(0))\right] \cdot \frac{v^{2}}{1+\frac{\alpha}{(1-\alpha)} v}\\ = \frac{v T(v)-v^{3} f(0)}{1-\alpha(1-v)}\\ E\left[D_{t}^{\alpha} f(t)\right] = \frac{v E[f(t)]-v^{3} f(0)}{1-\alpha(1-v)}. \end{array}$

(2.5)

Prabhakar introduced the generalized Mittag-Leffler function ^[22,23], in the following form

$E_{\sigma , \eta }^\delta (z) = \sum\nolimits_{r = 0}^\infty {} \frac{{\mathit{\Gamma} (\delta + k)}}{{\mathit{\Gamma} (\delta )\mathit{\Gamma} (\sigma k + \eta )}}\frac{{{z^r}}}{{r!}}$

(2.6)

Where $\sigma, \delta, \eta \in \mathbb{C}$ and $\mathbb{R}(\sigma)>0$

Another important result which will be used in this study ^[24], is given by

$t^{\eta-1} E_{\sigma, \eta}^{\delta}\left(\omega t^{\sigma}\right) = E_{\sigma, \eta, \omega}^{\delta}(t), \sigma, \delta, \eta, \omega \in \mathbb{C}, t \in \mathbb{R} \text { with } \mathbb{R}(\eta), \mathbb{R}(\sigma) \gt 0.$

(2.7)

The well known Prabhakar integral is expressed in the similar way, replacing kernel by function and is defined as follows ^[24,25].

The Hilfer-Prabhakar derivative of $g\left(t \right)$ of order $\mu$ denoted by $D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} g(t)$ and defined as

$D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} g(t) = \left(E_{\sigma, \nu(1-\mu), \omega, 0^{+}}^{-\gamma \nu}+\frac{d}{d t}\left(E_{\sigma, (1-\nu)(1-\mu), \omega, 0^{+}}^{-\gamma(1-\nu)}g\right)\right)(t),$

(2.8)

Where $\mu \in (0, 1), v \in [0, 1]$ and $\text{ }\gamma, \omega \in \mathbb{R}, \sigma >0$ , and $E_{\sigma, 0, \omega, 0^{+}}^{0}g = g$ .

The Elzaki transform of the Hilfer-Prabhakar derivative of fractional order is given by

$E\left( {E_{\sigma ,v(1 - \mu ),\omega ,{0^ + }}^{ - \gamma v}\frac{d}{{dt}}\left( {E_{\sigma ,(1 - v)(1 - \mu ),\omega ,{0^ + }}^{ - \gamma (1 - v)}g} \right)} \right)(p) \\ = p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma} E[g](p)-p^{v(1-\mu)+1}\left(1-\omega p^{\sigma}\right)^{\gamma v}\left[E_{\sigma, (1-v)(1-\mu), \omega, 0^{+}}^{-\gamma(1-v)} g(t)\right]_{t = 0^{+}}$

(2.9)

Proof: By taking Elzaki transform of Hilfer-Prabhakar fractional derivative and using (2.7), (2.8) and convolution theorem of Elzaki transform, we have

$\begin{array}{l} {E\left(D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} g(t)\right)(p)} \\ {\quad = \frac{1}{p} E\left[t^{\nu(1-\mu)-1} E_{\sigma, \nu(1-\mu)}^{-\gamma \nu}\left(\omega t^{\sigma}\right)\right](p) \cdot E\left[\frac{d}{d t}\left(E_{\sigma, (1-\nu)(1-\mu), \omega, 0^{+}}g\right)\right](p)}, \\ = p^{v(1-\mu)}\left(1-\omega p^{\sigma}\right)^{\gamma v} \cdot E\left[\frac{d}{d t}\left(E_{\sigma, (1-v)(1-\mu), \omega, 0^{+}} g\right)\right](p)\\ = {{p}^{v(1-\mu )}}{{\left( 1-\omega {{p}^{\sigma }} \right)}^{\gamma v}}{{\left[ {{p}^{v\mu -v-\mu }}{{\left( 1-\omega {{p}^{\sigma }} \right)}^{\gamma (1-v)}}E[g](p)-p\left( E_{\sigma , (1-v)(1-\mu ), \omega , {{0}^{+}}}^{-\gamma (1-v)}g \right) \right.}_{t = {{0}^{+}}}}\left. {} \right] \end{array}$

on simplification, we get the required result (2.9).

One more formula for Elzaki transform is given by

$E\left[t^{\eta-1} E_{\sigma, \eta}^{\delta}\left(\omega t^{\sigma}\right)\right] = p^{\eta+1}\left[1-\omega p^{\sigma}\right]^{-\delta}$

(2.10)

Fourier sine transform of function $f\left(x \right)$ is defined as

${{F}_{sin e}}\{f(x)\} = {{{\hat{f}}}_{{sine}}}(k) = \sqrt{2/\pi }\int_{0}^{\infty }{f}(x)sin e(kx)dx$

(2.11)

3. Fractional diffusion equations

In this portion, we analyze the FDE (fractional diffusion equation) in one and two-dimensional spaces. The phenomena of diffusion equation ^[5,25] is express in the following from

$\frac{\partial u(x, t)}{\partial t} = k^{2} \frac{\partial^{2} u(x, t)}{\partial x^{2}}$

(3.1)

where $k^{2} = \frac{K}{\rho c_{p}}$ .

● K entitled the thermal conductivity, $\rho$ for the specific heat, ${{C}_{p}}$ for the density of the material, $u$ for the temperature distribution of the material.

The FDE characterise by the Hilfer-Prabhakar fractional derivatives is demonstrate in one-dimensional space by the following equation ^[4,25,26,27]

$D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} u(x, t) = k^{2} \frac{\partial^{2} u(x, t)}{\partial x^{2}}, k^{2} \text { means diffusion coefficient }$

(3.2)

Where $D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu }$ represents the Hilfer-Prabhakar fractional derivatives operator defined by

$D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} f(t) = \left(E_{\sigma, \nu(1-\mu), \omega, 0+}^{-\gamma v} \frac{d}{d t}\left(E_{\sigma, (1-\nu)(1-\mu), \omega, 0+}^{-\gamma(1-\nu)} f\right)\right)(t)$

(3.3)

where $\gamma, \omega \in \mathbb{R}, \sigma>0$ and $E_{\sigma, 0, \omega, 0+}^{0} f = f$ . In this article, we used Dirichlet boundary conditions:

● $u(x, 0) = 0$ when $x>0$ ,

● $u(0, t) = 1$ when $t>0$

In two-dimensional space the fractional diffusion equation is given by ^[26]

$D_{\sigma, \omega, 0+}^{\gamma, \mu, v} u(x, y, t) = k^{2}\left\{\frac{\partial^{2} u(x, y, t)}{\partial x^{2}}+\frac{\partial^{2} u(x, y, t)}{\partial y^{2}}\right\}$

(3.4)

with B.C.

● $u(x, y, 0) = 0$ when $x, y>0$

● $u(0, y, t) = u(x, 0, t) = 1$ when $t>0$

The boundary conditions play a vital role in the integral method. Note that, when the boundary conditions change, the form of the analytic solution will changes also. There are many methods exist in literature to obtained the analytical solution of the fractional diffusion equation as the Elzaki transform, as the FST (Fourier sine transform) ^[28] and many more. This paper proposes an integral method consisting of applying both the Elzaki transform and Fourier sine transform.

4. Analytical solution of fractional diffusion equation

In this portion, we look into to get the analytical solution of the fractional diffusion equation in 1-dimsional space defined by (3.2), under the Dirichlet boundary conditions defined in a section 3. We can say that the initially temperature of the material is zero and the temp. of the plate (for all is keep up constant U₀ = 1. For more detail see in ^[15,17]. In this article, we assume the following integral method (see in ^[28]), described as follows:

● Using the FST

● Using the Elzaki transform

Figure 1. Fractional diffusion model.

DownLoad: Full-Size Img PowerPoint

➢ Using IET (inverse Elzaki transform)

➢ Using IFST (inverse Fourier sine transform)

To solve the Eq (3.2), first we applying the FST and then multiply by $\frac{2}{\text{ }\!\!\pi\!\!\text{ }}\sin \text{ }\!\!\eta\!\!\text{ x}$ and integrating it between 0 to , we arrive at:

$\begin{array}{c} {D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} u_{s}(\eta, t) = k^{2}\left\{\frac{2}{\pi} \eta u_{s}(0, t)-\eta^{2} u_{s}(\eta, t)\right\}} \\ {D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} u_{s}(\eta, t) = \frac{2 k^{2} \eta}{\pi}-k^{2} \eta^{2} u_{s}(\eta, t)} \end{array}$

where ${{u}_{s}}(\eta, t)$ denote the Fourier sine transform of $u(x,t)$ . After rearranging, we arrive at the following FDE defined as

$D_{\sigma, \omega, 0+}^{\gamma, \mu, \nu} u_{s}(\eta, t)+k^{2} \eta^{2} u_{s}(\eta, t) = \frac{2 k^{2} \eta}{\pi}$

(4.1)

applying the Elzaki transform to both sides of Eq (4.1), and by using (2.9), we have

$\begin{array}{l} &p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma} \overline{u_{s}}(\eta, p)-p^{\nu(1-\mu)+1}\left(1-\omega p^{\sigma}\right)^{\gamma v} f(\eta)+k^{2} \eta^{2} \overline{u_{s}}(\eta, p) = \frac{2 k^{2} \eta p^{2}}{\pi}\\ &\overline{u_{s}}(\eta, p)\left[p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}+k^{2} \eta^{2}\right] = \frac{2 k^{2} \eta p^{2}}{\pi}+p^{v(1-\mu)+1}\left(1-\omega p^{\sigma}\right)^{\gamma v} f(\eta) \end{array}$

$\begin{array}{l} &\overline{u_{s}}(\eta, p) = \frac{2 k^{2} \eta p^{2}}{\pi\left[p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}+k^{2} \eta^{2}\right]}+\frac{p^{v(1-\mu)+1}\left(1-\omega p^{\sigma}\right)^{\gamma v} f(\eta)}{\left[p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}+k^{2} \eta^{2}\right]}\\ &\overline{u_{s}}(\eta, p) = \frac{2 k^{2} \eta p^{2}}{\pi p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}\left[1+\frac{k^{2} \eta^{2}}{p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}}\right]}+\frac{p^{\nu(1-\mu)+1}\left(1-\omega p^{\sigma}\right)^{\gamma \nu} f(\eta)}{p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}\left[1+\frac{k^{2} \eta^{2}}{p^{-\mu}\left(1-\omega p^{\sigma}\right) \gamma}\right]} \end{array}$

$\begin{array}{l} &\overline{u_{s}}(\eta, p) = \frac{2 k^{2} \eta}{\pi} p^{\mu+2}\left(1-\omega p^{\sigma}\right)^{-\gamma}\left[1+\frac{k^{2} \eta^{2}}{p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}}\right]^{-1}\\ &+p^{v(1-\mu)+\mu+1}\left(1-\omega p^{\sigma}\right)^{\gamma v-\gamma} f(\eta)\left[1+\frac{k^{2} \eta^{2}}{p^{-\mu}\left(1-\omega p^{\sigma}\right)^{\gamma}}\right]^{-1} \end{array}$

$\begin{array}{l} \overline {{u_s}} (\eta , p) = \frac{2}{\pi }\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {k^{2n + 2}}{\eta ^{2n + 1}}{p^{(\mu (n + 1) + 1) + 1}}{\left( {1 - \omega {p^\sigma }} \right)^{ - (n + 1)\gamma }}\\ + \sum\nolimits_{n = 0}^\infty {} {( - 1)^n}{k^{2n}}{\eta ^{2n}}{p^{(\nu (1 - \mu ) + (n + 1)\mu ) + 1}}{\left( {1 - \omega {p^\sigma }} \right)^{ - \gamma ((n + 1) - v)}}f(\eta ) \end{array}$

(4.2)

Where $\overline {{u_s}} (\eta, p)$ denoted the Elzaki transform of $u_{s}(\eta, t)$ . Now, we applying the inverse of the Elzaki transform to both sides of Eq (4.2) and using the Eq (2.10) as follows:

$\begin{array}{l} {u_s}(\eta , t) = \frac{2}{\pi }\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {k^{2n + 2}}{\eta ^{2n + 1}}{t^{\mu (n + 1)}}E_{\sigma , \mu (n + 1) + 1}^{(n + 1)\gamma }\left( {\omega {t^\sigma }} \right)\\ + \sum\nolimits_{n = 0}^\infty {} {( - 1)^n}{k^{2n}}{\eta ^{2n}}{t^{\nu (1 - \mu ) + (n + 1)\mu - 1}}E_{\sigma , (v(1 - \mu ) + (n + 1)\mu }^{\gamma ((n + 1) - v)}\left( {\omega {t^\sigma }} \right)f(\eta ) \end{array}$

(4.3)

Finally, to obtained the analytical solution of Eq (3.2), we applying the inverse of the Fourier sine transform to both sides of Eq (4.3), then we arrive at the following result

$\begin{array}{l} u(x, t) = \frac{2}{\pi }\int_0^\infty {\sin } \eta x\left( {\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {k^{2n + 2}}{\eta ^{2n + 1}}{t^{\mu (n + 1)}}E_{\sigma , \mu (n + 1) + 1}^{(n + 1)\gamma }\left( {\omega {t^\sigma }} \right)} \right)d\eta \\ + \frac{2}{\pi }\int_0^\infty f (\eta )\sin \eta x\left( {\sum\nolimits_{n = 0}^\infty {} {{( - 1)}^n}{k^{2n}}{\eta ^{2n}}{t^{\nu (1 - \mu ) + (n + 1)\mu - 1}}E_{\sigma , (\nu (1 - \mu ) + (n + 1)\mu )}^{\gamma ((n + 1) - \nu )}\left( {\omega {t^\sigma }} \right)} \right)d\eta . \end{array}$

(4.4)

This is complete analytical solution of the fractional diffusion Eq (3.2).

5. Analytical solution of the fractional diffusion equation in "two dimensional spaces"

In this portion, we find the analytical solution of the FDE in two-dimensional space defined by (3.4), with same boundary conditions.

We follow the same process as we done in the section 4. Now applying the Fourier sine transform and multiplying Eq (3.4) by $\frac{2}{\pi} \sin \omega x \sin \eta y$ and integrating it between 0 to $\infty$ w.r.to $x$ and $y$ , we arrive at

$D_{\sigma, \tau, 0+}^{\gamma, \mu, \nu} u_{s}(\omega, \eta, t) = k^{2}\left\{\frac{2\left(\omega^{2}+\eta^{2}\right)}{\pi \omega \eta} u_{s}(0, 0, t)-\left(\omega^{2}+\eta^{2}\right) u_{s}(\omega, \eta, t)\right\} \\ D_{\sigma, \tau, 0+}^{\gamma, \mu, \nu} u_{s}(\omega, \eta, t) = \frac{2 k^{2}\left(\omega^{2}+\eta^{2}\right)}{\pi \omega \eta}-k^{2}\left(\omega^{2}+\eta^{2}\right) u_{s}(\omega, \eta, t)$

where $u_{s}(\omega, \eta, t)$ denotes the Fourier sine transform of $u(x, y, t)$ . After Rearranging, the fractional diffusion equation defined as

$D_{\sigma, \tau, 0+}^{\gamma, \mu, \nu} u_{s}(\omega, \eta, t)+k^{2}\left(\omega^{2}+\eta^{2}\right) u_{s}(\omega, \eta, t) = \frac{2 k^{2}\left(\omega^{2}+\eta^{2}\right)}{\pi \omega \eta}$

(5.1)

We apply the Elzaki transform to both sides of Eq (5.1) and by using (2.9), we have

$p^{-\mu}\left(1-\tau p^{\sigma}\right)^{\gamma} \overline{u_{s}}(\omega, \eta, p)-p^{\nu(1-\mu)+1}\left(1-\tau p^{\sigma}\right)^{\gamma v} f(\omega, \eta)+k^{2}\left(\omega^{2}+\eta^{2}\right) \overline{u_{s}}(\omega, \eta, p)\\ = \frac{2 k^{2}\left(\omega^{2}+\eta^{2}\right) p^{2}}{\pi \omega \eta}$

$\overline{u_{s}}(\omega, \eta, p)\left[p^{-\mu}\left(1-\tau p^{\sigma}\right)^{\gamma}+k^{2}\left(\omega^{2}+\eta^{2}\right)\right]\\ = \frac{2 k^{2}\left(\omega^{2}+\eta^{2}\right) p^{2}}{\pi \omega \eta}+p^{\nu(1-\mu)+1}\left(1-\tau p^{\sigma}\right)^{\gamma \nu} f(\omega, \eta)$

$\overline{u_{s}}(\omega, \eta, p) = \frac{2 k^{2}\left(\omega^{2}+\eta^{2}\right) p^{2}}{\pi \omega \eta\left[p^{-\mu}\left(1-\tau p^{\sigma}\right)^{\gamma}+k^{2}\left(\omega^{2}+\eta^{2}\right)\right]}+\frac{p^{\nu(1-\mu)+1}\left(1-\tau p^{\sigma}\right)^{\gamma \nu} f(\omega, \eta)}{\left[p^{-\mu}\left(1-\tau p^{\sigma}\right)^{\gamma}+k^{2}\left(\omega^{2}+\eta^{2}\right)\right]}$

$\begin{array}{l} \overline {{u_s}} (\omega , \eta , p) = \frac{2}{{\pi \omega \eta }}\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {k^{2n + 2}}{\left( {{\omega ^2} + {\eta ^2}} \right)^{n + 1}}{p^{(\mu (n + 1) + 1) + 1}}{\left( {1 - \tau {p^\sigma }} \right)^{ - (n + 1)\gamma }}\\ + \sum\nolimits_{n = 0}^\infty {} {( - 1)^n}{k^{2n}}{\left( {{\omega ^2} + {\eta ^2}} \right)^n}{p^{(\nu (1 - \mu ) + (n + 1)\mu ) + 1}}{\left( {1 - \tau {p^\sigma }} \right)^{ - \gamma ((n + 1) - v)}}f(\omega , \eta ) \end{array}$

(5.2)

Where $\overline {{u_s}} (\omega, \eta, p)$ denoted the Elzaki transform of $u_{s}(\omega, \eta, t)$ . The third step of the solution, we applying the inverse of the Elzaki transform to both sides of Eq (5.2) and using the Eq (2.10) as follows:

$\begin{array}{l} {u_s}(\omega , \eta , t) = \frac{2}{{\pi \omega \eta }}\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {k^{2n + 2}}{\left( {{\omega ^2} + {\eta ^2}} \right)^{n + 1}}{t^{\mu (n + 1)}}E_{\sigma , \mu (n + 1) + 1}^{(n + 1)\gamma }\left( {\tau {t^\sigma }} \right)\\ + \sum\nolimits_{n = 0}^\infty {} {( - 1)^n}{k^{2n}}{\left( {{\omega ^2} + {\eta ^2}} \right)^n}{t^{\nu (1 - \mu ) + (n + 1)\mu - 1}}E_{\sigma , (\nu (1 - \mu ) + (n + 1)\mu )}^{\gamma ((n + 1) - v)}\left( {\tau {t^\sigma }} \right)f(\omega , \eta ) \end{array}$

(5.3)

Now, we apply the inverse Fourier sine transform to both sides of Eq (5.3)

$u(x, y, t) = \frac{4}{\pi^{2}} \int_{0}^{\infty} \frac{\sin \omega x}{\omega} \int_{0}^{\infty} \frac{\sin \eta y}{\eta}\left\{\sum\limits_{n = 0}^{\infty}(-1)^{n} k^{2 n+2}\left(\omega^{2}+\eta^{2}\right)^{n+1} t^{\mu(n+1)} E_{\sigma, \mu(n+1)+1}^{(n+1) \gamma}\left(\tau t^{\sigma}\right)\right\} d \omega d \eta \\ \begin{array}{*{20}{c}} { + \frac{4}{{{\pi ^2}}}\int\limits_0^\infty {} \int\limits_0^\infty {} \sin \omega x\sin \eta yf(\omega , \eta )}\\ { \cdot \left\{ {\sum\nolimits_{n = 0}^\infty {} {{( - 1)}^n}{k^{2n}}{{\left( {{\omega ^2} + {\eta ^2}} \right)}^n}{t^{\nu (1 - \mu ) + (n + 1)\mu - 1}}E_{\sigma , (\nu (1 - \mu ) + (n + 1)\mu }^{\gamma ((n + 1) - \nu )}\left( {\tau {t^\sigma }} \right)} \right\}d\omega d\eta } \end{array}$

(5.4)

This is complete analytical solution of the fractional diffusion Eq (3.4).

6. Analytical solution of the Cattaneo-Hristov diffusion equation

Hristov in ^[7,8] described the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time fractional derivative. Diffusion phenomena, of heat or mass are generally express as ^[5]:

$\rho C_{p} \frac{\partial T}{\partial t} = -\frac{\partial q}{\partial x} ; \quad q(x, t) = -k \frac{\partial T(x, t)}{\partial x} \Rightarrow \rho C_{p} \frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^{2}}$

(6.1)

where $q(x, t)$ the flux of heat and it is express by the following relationship

$q(x, t) = -\int_{-\infty}^{t} R(x, t) \nabla T(x, t-s) d s$

(6.2)

where $R(x, t)$ is space independent it can be denoted by the Jeffrey kernel $R(t) = exp (-(t-s) / \tau)$ where $\tau$ is relaxation time ^[5,9]. The energy balance produces the Cattaneo equation and defined as ^[5]:

$\frac{\partial T(x, t)}{\partial t} = \frac{k_{2}}{\mathrm{T} \rho c_{p}} \int_{0}^{t} exp (-(t-s) / \tau) \frac{\partial T(x, s)}{\partial x} d s$

(6.3)

In continuation of the Eq (6.3), the Jeffrey type intero-differential equation ^[5] express in the following form

$\frac{\partial T(x, t)}{\partial t} = \frac{k_{1}}{\rho c_{p}} \frac{\partial^{2} T(x, t)}{\partial x^{2}}+\frac{k_{2}}{T \rho c_{p}} \int_{-\infty}^{t} exp (-(t-s) / \tau) \frac{\partial^{2} T(x, s)}{\partial x^{2}} d s$

(6.4)

In the end, by applying the concept of the Caputo-Fabrizio fractional derivative recently introduced in ^[2], Hristov comes to the complete Cattaneo-Hristov diffusion equation ^[5,6] and defined as

$\frac{\partial T(x, t)}{\partial t} = a_{1} \frac{\partial^{2} T(x, t)}{\partial x^{2}}+a_{2}(1-\alpha)_{0}^{C F} D_{t}^{\alpha}\left(\frac{\partial^{2} T(x, t)}{\partial x^{2}}\right)$

(6.5)

where T denote the temperature distribution, ${a_1} = \frac{{{k_1}}}{{\rho {c_p}}}$ and ${a_2} = \frac{{{k_2}}}{{\rho {c_p}}}$ with ${\rm{ }}\rho = {\rm{ }}const$ , ${C_p} = constan t$ . The constant ${k_1}$ and ${k_2}$ is effective thermal conductivity and the elastic conductivity. denotes the Caputo-Fabrizio fractional derivative, see in ^[2].The boundary conditions defined in the following form

● $u(x, 0) = 0$ for ${\rm{ }}x > 0$ ,

● $u(0, t) = 1$ for ${\rm{ }}t > 0$

The Eq (6.5) is known as the entire Cattaneo-Hristov equation of transition heat diffusion equation and for detail see ^[10].

The second term of the Cattaneo-Hristov equation is express as

$\frac{\partial T(x, t)}{\partial t} = a_{2}(1-\alpha)_{0}^{C F} D_{t}^{\alpha}\left(\frac{\partial^{2} T(x, t)}{\partial x^{2}}\right)$

(6.6)

is known as the elastic part of the heat diffusion equation process and it was subject of investigations done by Koca et al. in ^[9].

In this part, we explore to find the analytical solution of the complete Cattaneo-Hristov diffusion Eq (6.5) with Caputo-Fabrizio fractional derivative by applying Elzaki transform. The boundary conditions considered in this paper are particular cases which we can obtain with Cattaneo-Hristov model of diffusion. All these results obtained in this part can be change when the boundary conditions are changes. Now, before applying the Fourier sine transform and the Elzaki transform, we recall the Elzaki transform of the Caputo-Fabrizio fractional derivative given by

$E\left[ {\begin{array}{*{20}{c}} {CF}\\ 0 \end{array}D_t^\alpha f(t)} \right] = \frac{{vE[f(t)] - {v^3}f(0)}}{{1 - \alpha (1 - v)}}$

(6.7)

To obtained the solution of the CCHD equation first we multiply the Eq (6.5) by $\frac{2}{\pi} \sin \omega x$ and integrating it between limit 0 to $\infty$ ; we arrive at

$\begin{array}{l} \frac{{\partial {{\rm{T}}_{\rm{s}}}(\omega , {\rm{t}})}}{{\partial {\rm{t}}}} = {{\rm{a}}_1}\left\{ {\frac{2}{\pi }\omega {{\rm{T}}_{\rm{s}}}(0, {\rm{t}}) - {\omega ^2}{{\rm{T}}_{\rm{s}}}(\omega , {\rm{t}})} \right\} + {{\rm{a}}_2}(1 - \alpha )\begin{array}{*{20}{c}} {{\rm{CF}}}\\ {\rm{0}} \end{array}D_{\rm{t}}^{\rm{ \mathsf{ α} }}\left\{ {\frac{2}{\pi }\omega {{\rm{T}}_{\rm{s}}}(0, {\rm{t}}) - {\omega ^2}{{\rm{T}}_{\rm{s}}}(\omega , {\rm{t}})} \right\}\\ = {a_1}\left\{ {\frac{2}{\pi }\omega - {\omega ^2}{T_s}(\omega , t)} \right\} + {a_2}{\omega ^2}(1 - \alpha )_0^{CF}D_t^\alpha {T_s}(\omega , t) \end{array}$

(6.8)

where $T_{s}(\omega, t) = F_{s}[\mathrm{T}(\mathrm{x}; \mathrm{t})]$ .

Now, taking Elzaki transform to both sides of Eq (6.8), we arrive at

$\frac{\bar{T}_{s}(\omega, t)}{v}-v T_{s}(\omega, 0)+a_{1} \omega^{2} \bar{T}_{s}(\omega, t)+a_{2} \omega^{2}(1-\alpha)\left\{\frac{v \bar{T}_{s}(\omega, t)-v^{3} T_{s}(\omega, 0)}{1-\alpha(1-v)}\right\} = \frac{2 a_{1} \omega v^{2}}{\pi}\\ \bar{T}_{s}(\omega, t)\left[\frac{1}{v}+a_{1} \omega^{2}+\frac{a_{2} \omega^{2}(1-\alpha) v}{1-\alpha(1-v)}\right] = \frac{2 a_{1} \omega v^{2}}{\pi}\\ \bar{T}_{s}(\omega, t)\left[\frac{(1-\alpha(1-v))+a_{1} \omega^{2} v(1-\alpha(1-v))+a_{2} \omega^{2}(1-\alpha) v^{2}}{v(1-\alpha(1-v))}\right] = \frac{2 a_{1} \omega v^{2}}{\pi}\\ \bar{T}_{s}(\omega, t) = \frac{2 a_{1} \omega v^{3}}{\pi} \frac{(1-\alpha+\alpha v)}{\left[(1-\alpha+\alpha v)+a_{1} \omega^{2} v(1-\alpha+\alpha v)+a_{2} \omega^{2}(1-\alpha) v^{2}\right]}$

(6.9)

where $\bar{T}_{s}(\omega, t)$ denotes the Elzaki transform of ${T_s}(\omega, t)$ . Let that $\lambda = \frac{\alpha }{{1 - \alpha }}$ with $\alpha \ne 1$ and then Eq (6.9) can be rewritten as follows

$\begin{array}{l} \bar{T}_{s}(\omega, t) & = \frac{2 a_{1} \omega v^{3}(1+\lambda v)}{\pi\left[(1+\lambda v)+a_{1} \omega^{2} v(1+\lambda v)+a_{2} \omega^{2} v^{2}\right]} \\ \bar{T}_{s}(\omega, t) & = \frac{2 a_{1} \omega v^{3}(1+\lambda v)}{\pi\left[1+\left(\lambda+a_{1} \omega^{2}\right) v+\left(a_{1} \omega^{2} \lambda+a_{2} \omega^{2}\right) v^{2}\right]} \end{array}$

Put $\left(\lambda+a_{1} \omega^{2}\right) = -\theta_{1}, \left(a_{1} \omega^{2} \lambda+a_{2} \omega^{2}\right) = \theta_{2}$ ,

$\begin{array}{l} &\bar{T}_{s}(\omega, t) = \frac{2 a_{1} \omega v^{3}(1+\lambda v)}{\pi\left[1-\theta_{1} v+\theta_{2} v^{2}\right]}\\ &\bar{T}_{s}(\omega, t) = \frac{2 a_{1} \omega\left(v^{3}+\lambda v^{4}\right)}{\pi\left(1-\theta_{1} v\right)}\left[1+\frac{\theta_{2} v^{2}}{1-\theta_{1} v}\right]^{-1} \end{array}$

$\bar{T}_{s}(\omega, t) = \frac{2 \omega a_{1}}{\pi}\left[\sum\limits_{n = 0}^{\infty}(-1)^{n}\left(\theta_{2}\right)^{n} \frac{v^{2 n+3}}{\left(1-\theta_{1} v\right)^{n+1}}+\sum\limits_{n = 0}^{\infty}(-1)^{n} \lambda\left(\theta_{2}\right)^{n} \frac{v^{2 n+4}}{\left(1-\theta_{1} v\right)^{n+1}}\right]$

$\begin{array}{l} {{\bar T}_s}(\omega , t) = \frac{2}{\pi }\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {a_1}\omega {\left( {{\theta _2}} \right)^n}{v^{(2n + 2) + 1}}{\left( {1 - {\theta _1}v} \right)^{ - (n + 1)}}\\ + \frac{2}{\pi }\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {a_1}\omega \lambda {\left( {{\theta _2}} \right)^n}{v^{(2n + 3) + 1}}{\left( {1 - {\theta _1}v} \right)^{ - (n + 1)}} \end{array}$

Appling the inverse of Elzaki transform and using Eq (2.7), we get

$\begin{array}{l} {T_s}(\omega , t) = \frac{2}{\pi }\sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {a_1}\omega {\left( {{\theta _2}} \right)^n}{t^{(2n + 1)}}E_{1, (2n + 2)}^{(n + 1)}\left( {{\theta _1}t} \right)\\ + \frac{2}{\pi }\sum\nolimits_{n = 0}^\infty {} {( - 1)^n}{a_1}\omega \lambda {\left( {{\theta _2}} \right)^n}{t^{(2n + 2)}}E_{1, (2n + 3)}^{(n + 1)}\left( {{\theta _1}t} \right) \end{array}$

(6.10)

Now, we taking the inverse of Fourier sine transform and we obtained the solution of the Cattaneo-Hristov diffusion equation

$\begin{array}{l} T(x, t) & = \frac{2}{\pi} \int_{0}^{\infty} \sin\limits \omega x \cdot \omega\left[\sum\limits_{n = 0}^{\infty}(-1)^{n} a_{1}\left(\theta_{2}\right)^{n} t^{(2 n+1)} E_{1, (2 n+2)}^{(n+1)}\left(\theta_{1} t\right)\right] d \omega \\ &+\frac{2}{\pi} \int_{0}^{\infty} \sin\limits \omega x \cdot \omega\left[\sum\limits_{n = 0}^{\infty}(-1)^{n} a_{1} \lambda\left(\theta_{2}\right)^{n} t^{(2 n+2)} E_{1, (2 n+3)}^{(n+1)}\left(\theta_{1} t\right)\right] d \omega \end{array}$

This is complete solution of the Cattaneo-Hristov diffusion Eq (6.5).

7. Conclusions

In this article, we explore the complete Cattaneo-Hristov diffusion equation and fractional diffusion equations in one and two dimensional spaces and find their analytic solution under the defined boundary conditions by using the Elzaki transform. We also find the solution of fractional diffusion equation associated with Hilfer-Prabhakar derivative in terms of Mittag-Leffler function for both one and two dimensional space equations. In this article, we establish new results of Elzaki transform of Caputo-Fabrizio and Hilfer-Prabhakar derivative which will be very helpful to find the analytical solution of various fractional differential equations.

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

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Abstract

1. Introduction

2. Definitions

3. Fractional diffusion equations

4. Analytical solution of fractional diffusion equation

5. Analytical solution of the fractional diffusion equation in "two dimensional spaces"

6. Analytical solution of the Cattaneo-Hristov diffusion equation

7. Conclusions

Conflict of interest

References

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AIMS Mathematics

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

Related Papers:

Abstract

1. Introduction

2. Definitions

3. Fractional diffusion equations

4. Analytical solution of fractional diffusion equation

5. Analytical solution of the fractional diffusion equation in "two dimensional spaces"

6. Analytical solution of the Cattaneo-Hristov diffusion equation

7. Conclusions

Conflict of interest

References

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