A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative

Yudhveer Singh; Devendra Kumar; Kanak Modi; Vinod Gill; Yudhveer Singh; Devendra Kumar; Kanak Modi; Vinod Gill

doi:10.3934/math.2020057

AIMS Mathematics

2020, Volume 5, Issue 2: 843-855. doi: 10.3934/math.2020057

Previous Article Next Article

Research article Special Issues

A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative

1.
Amity Institute of information Technology, Amity University, Rajasthan, Jaipur-303002, India
2.
Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India
3.
Department of Mathematics, Amity University, Rajasthan, Jaipur-303002, India
4.
Department of Mathematics, Govt. P. G. College, Hisar, Haryana-125001, India

Received: 09 October 2019 Accepted: 14 December 2019 Published: 31 December 2019
MSC : 26A33, 42A38, 76R50

In the present article, we investigate complete Cattaneo-Hristov diffusion (CCHD) equation and fractional diffusion equation in one and two dimensional spaces and find their analytic solution by using Elzaki transform technique under the Dirichlet boundary conditions. The fractional diffusion equation describe by the Hilfer-Prabhakar derivative and established the solution in one and two dimensional spaces by using Elzaki and Fourier Sine transform in terms of Mittag-Leffler function. In this paper, we also establish new results such as Elzaki transform of Caputo-Fabrizio and Hilfer-Prabhakar derivative which will be very helpful to find the analytical solution fractional differential equations.

Keywords:

Citation: Yudhveer Singh, Devendra Kumar, Kanak Modi, Vinod Gill. A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative[J]. AIMS Mathematics, 2020, 5(2): 843-855. doi: 10.3934/math.2020057

Related Papers:

[1]	Xiaoyong Xu, Fengying Zhou . Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative. AIMS Mathematics, 2023, 8(2): 2736-2762. doi: 10.3934/math.2023144
[2]	Aslı Alkan, Halil Anaç . A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative. AIMS Mathematics, 2024, 9(10): 27979-27997. doi: 10.3934/math.20241358
[3]	Alessandra Jannelli, Maria Paola Speciale . On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics, 2021, 6(8): 9109-9125. doi: 10.3934/math.2021529
[4]	Junseok Kim . A normalized Caputo–Fabrizio fractional diffusion equation. AIMS Mathematics, 2025, 10(3): 6195-6208. doi: 10.3934/math.2025282
[5]	Ritu Agarwal, Mahaveer Prasad Yadav, Dumitru Baleanu, S. D. Purohit . Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative. AIMS Mathematics, 2020, 5(2): 1062-1073. doi: 10.3934/math.2020074
[6]	Muhammed Naeem, Noufe H. Aljahdaly, Rasool Shah, Wajaree Weera . The study of fractional-order convection-reaction-diffusion equation via an Elzake Atangana-Baleanu operator. AIMS Mathematics, 2022, 7(10): 18080-18098. doi: 10.3934/math.2022995
[7]	Chaeyoung Lee, Yunjae Nam, Minjoon Bang, Seokjun Ham, Junseok Kim . Numerical investigation of the dynamics for a normalized time-fractional diffusion equation. AIMS Mathematics, 2024, 9(10): 26671-26687. doi: 10.3934/math.20241297
[8]	Krunal B. Kachhia, Jyotindra C. Prajapati . Generalized iterative method for the solution of linear and nonlinear fractional differential equations with composite fractional derivative operator. AIMS Mathematics, 2020, 5(4): 2888-2898. doi: 10.3934/math.2020186
[9]	Sami Ul Haq, Saeed Ullah Jan, Syed Inayat Ali Shah, Ilyas Khan, Jagdev Singh . Heat and mass transfer of fractional second grade fluid with slippage and ramped wall temperature using Caputo-Fabrizio fractional derivative approach. AIMS Mathematics, 2020, 5(4): 3056-3088. doi: 10.3934/math.2020198
[10]	Yumei Chen, Jiajie Zhang, Chao Pan . Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials. AIMS Mathematics, 2022, 7(8): 15612-15632. doi: 10.3934/math.2022855

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.

References

[1]	R. Garra, R. Gorenflo, F. polito, et al. Hilfer-Prabhakar derivatives and some applications. Appl. Math. Comput., 242 (2014), 576-589.
[2]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1 (2015), 1-13.
[3]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1 (2015), 87-92.
[4]	J. Hristov, Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Spec. Top, 193 (2011), 229-243.
[5]	J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey's Kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci., 20 (2016), 757-762.
[6]	J. Hristov, Derivation of the fractional dodson equation and beyond: Transient di_usion with a non-singular memory and exponentially fadingout diffusivity. Progr. Fract. Differ. Appl., 3 (2017), 1-16.
[7]	J. Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation, Fractional Derivatives with Mittag-Leffler Kernel, Stud. Syst. Decis. Control, Springer, Cham, 194 (2019), 175-193.
[8]	J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory derivation of caputo-fabrizio space fractional derivative from cattaneo concept with jeffrey's kernel and analytical solutions, Therm. Sci., 21 (2017), 827-839.
[9]	I. Koca, A. Atangana, Solutions of cattaneo-hristov model of elastic heat diffusion with caputo-fabrizio and atangana-baleanu fractional derivatives, Therm. Sci., 21 (2017), 2299-2305.
[10]	B. S. T. Alkahtani, A. Atangana, A note on cattaneo-Hristov model with non-singular fading memory. Therm. Sci., 21 (2017), 1-7.
[11]	J. Hristov, Multiple integral-balance method basic idea and an example with mullins model of thermal grooving. Therm. Sci., 21 (2017), 1555-1560.
[12]	J. Hristov, The non-linear dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization. Math. Nat. Sci., 1 (2017), 1-17.
[13]	J. Hristov, Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. Math. Modell. Nat. Phenom., 13 (2018), 6.
[14]	J. Hristov, Integral-balance solution to nonlinear subdiffusion equation. Front. Fractional. Calculus., 1 (2017), 71-106.
[15]	J. Hristov, The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method. Therm. Sci., 22 (2018), 777-788.
[16]	J. Hristov, Integral balance approach to 1-D space-fractional diffusion models, Mathematical Methods in Engineering, Nonlinear Syst. Complex. Springer Cham., 23 (2019), 111-131.
[17]	J. Hristov, A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach, Mathematical Methods in Engineering. Nonlinear Syst. Complex. Springer Cham., 24 (2019), 153-174.
[18]	T. G. Myers, Optimal exponent heat balance and refined integral methods applied to stefan problems. Int. J. Heat Mass Transfer., 53 (2010), 1119-1127.
[19]	E. F. D. Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications. Chaos: An Interdiscip. J. Nonlinear Sci., 26 (2016), 084305.
[20]	N. Sene, Exponential form for Lyapunov function and stability analysis of the fractional differential equations. J. Math. Comput. Sci., 18 (2018), 388-397.
[21]	Tarig. M. Elzaki, The new integral transform "ELzaki Transform", Global J. Pure Appl Math., 7 (2011) 57-64.
[22]	R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics, New York: Springer-Verlag Press (1997), 223-276.
[23]	A. A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr Transfor. Spec Funct., 15 (2004), 31-49.
[24]	T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J., 19 (1971), 7-5.
[25]	I. Podlubny, Matrix approach to discrete fractional calculus ii: Partial fractional differential equations. J. Comput. Phy., 228 (2009), 3131-3153.
[26]	Y. Ma, F. Zhang, C. Li, The asymptotics of the solutions to the anomalous diffusion equations. Comput. Math. Appl., 66 (2013), 682-692.
[27]	V. Gill, J. Singh, Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Frontiers. Phy., 6 (2019), 1-6.
[28]	N. Sene, Stokes first problem for heated at plate with Atangana-Baleanu fractional derivative. Chaos Solitons Fractals, 117 (2018), 68-75.

Reader Comments

Your name:*

Email:*
© 2020 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2899) PDF downloads(544) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(1)

AIMS Mathematics

A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative

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

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative

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

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog