Solution of fractional Distributed Order Reaction-Diffusion Systems with Sumudu Transform

TL;DR

This paper develops an analytical solution for fractional distributed order reaction-diffusion equations using Laplace-Sumudu transform, advancing methods in fractional calculus.

Contribution

It introduces a novel approach combining Laplace and Sumudu transforms to solve fractional distributed order reaction-diffusion systems.

Findings

Analytical solutions derived for fractional distributed order reaction-diffusion equations.

Effective application of Laplace-Sumudu transform method.

Contributes to analytical techniques in fractional calculus.

Abstract

The solution of some fractional differential equations is the hottest topic in fractional calculus field. The fractional distributed order reaction-diffusion equation is the aim of this paper. By applying integral transform to solve this type of fractional differential equations, we have obtained the analytical solution by using Laplace-Sumudu transform.

Equations

$$\mathcal{L}\{f(x,t);s\}=\mathcal{F}(x,s)={\int }_0^{\infty }{e}^{-st}f(x,t)dt,(t>0),(x\in R)$$

$${\mathcal{L}}^{-1}\{\mathcal{F}(x,s)\}=f(x,t)=\frac{1}{2\pi i}{\int }_{\nu -i\infty }^{\nu +i\infty }{e}^{st}\mathcal{F}(x,s)ds,$$

$${}_0{I}_t^{\nu }f(x,t)$$

$${}_0{I}_t^{0}f(x,t)$$

$${}_0{D}_t^{\alpha }f(x,t)=\frac{d^m}{d{t}^m}\left({}_0{I}_t^{m-\alpha }f(x,t)\right),(t>0),(m=[\alpha +1])$$

$${}_{0}^{c}D_{t}^{\alpha}f(x,t)=\left\{\begin{array}[]{ll}{}_{0}I_{t}^{m-\alpha}f^{(m)}(x,t),&~{}~{}~{}\text{where}~{}~{}m-1<\alpha\leq m,Re(\alpha)>0,m\in N;\\ f^{(m)}(x,t),&~{}~{}~{}\text{where}~{}~{}\alpha=m.\end{array}\right..$$

$$\mathcal{L}{\{}_0{D}_t^{\alpha }f(x,t);s\}$$

$$\mathcal{L}{\{}_0^c{D}_t^{\alpha }f(x,t);s\}$$

$$A=\left\{f(t)|\exists M,{\tau }_1,{\tau }_2>0,\left|f(t)\right|<M{e}^{|t|/{\tau }_j},\text{if }t\in (-1{)}^j\times [0,\infty )\right\}$$

$$G(u)=S\left[f(t)\right]={\int }_0^{\infty }f(ut)e^{-t}dt,u\in (-{\tau }_1,{\tau }_2)$$

$$S^{-1}\left[u^{\gamma -1}(1-w{u}^{\beta }{)}^{-\delta }\right]=t^{\gamma -1}{E}_{\beta ,\gamma }^{\delta }(w{t}^{\beta })$$

$$S^{-1}\left[\frac{u^{-\rho }}{u^{-\alpha }+\lambda u^{-\beta }+b}\right]=\sum _{r=0}^{\infty }(-\lambda {)}^r{t}^{(\alpha -\beta )r+\alpha -\rho }{E}_{\alpha ,\alpha +(\alpha -\beta )r-\rho +1}^{r+1}(-b{t}^{\alpha })$$

$$S^{-1}\left[\frac{u^{-\rho }}{u^{-\alpha }+b}\right]=t^{\alpha -\rho }{E}_{\alpha ,\alpha -\rho +1}^1(-b{t}^{\alpha })$$

$$E_{\alpha ,\beta }^{\delta }(z)=\sum _{n=0}^{\infty }\frac{(\delta {)}_n{z}^n}{\Gamma (n\alpha +\beta )n!},(\alpha ,\beta ,\delta \in \mathbb{C},\Re (\alpha )>0,\Re (\beta )>0,\Re (\delta )>0)$$

$$S^{-1}\left[uS[\psi (t)]S[\varphi (t)]\right]=(\psi \ast \varphi )(t)={\int }_0^t\psi (\xi )\varphi (t-\xi )d\xi$$

$$S\left[{}_0{D}_t^{\alpha }f(x,t);u\right]=u^{-\alpha }\mathbb{F}(x,u)-\sum _{k=1}^m\frac{{\left[{}_0{D}_t^{\alpha -k}f(x,t)\right]}_{t=0}}{u^k},(m-1<\alpha \le m)$$

$$S\left[{}_0^C{D}_t^{\alpha }f(x,t);u\right]=u^{-\alpha }\mathbb{F}(x,u)-\sum _{k=0}^{m-1}\frac{1}{u^{\alpha -k}}{\left[\frac{{\partial }^{k}f(x,t)}{\partial t^k}\right]}_{t=0},(m-1<\alpha \le m)$$

$${}_0{D}_t^{\alpha }N(x,t)=\sum _{j=1}^n{\mu }_{jx}{D}_{{\theta }_j}^{{\gamma }_j}N(x,t)+\varphi (x,t)$$

$${\mu }_j>0,0<{\gamma }_j\le 2,j=1,2,...,n,\left|{\theta }_j\right|\le \underset{1\le j\le n}{\min }({\gamma }_j,2-{\gamma }_j),1<\alpha \le 2$$

$$\left(\begin{array}[]{c}{}_{0}D_{t}^{\alpha-1}N(x,t)\\ {}_{0}D_{t}^{\alpha-2}N(x,t)\\ \end{array}\right)_{t=0}=\left(\begin{array}[]{c}f(x)\\ g(x)\\ \end{array}\right),\,x\in\mathbb{R},\,\lim_{x\rightarrow\pm\infty}N(x,t)=0,t>0$$

$$N(x,t)$$

$$S\left[{}_0{D}_t^{\alpha }N(x,t);u\right]=S\left[\sum _{j=1}^n{\mu }_{jx}{D}_{{\theta }_j}^{{\gamma }_j}N(x,t);u\right]+S\left[\varphi (x,t);u\right]$$

$$u^{-\alpha }\mathbb{N}(x,u)-\left[\frac{{}_0{D}_t^{\alpha -1}N(x,0)}{u}\right]-\left[\frac{{}_0{D}_t^{\alpha -2}N(x,0)}{u^2}\right]=\sum _{j=1}^n{\mu }_{jx}{D}_{{\theta }_j}^{{\gamma }_j}\mathbb{N}(x,u)+\Phi (x,u)$$

$$u^{-\alpha }\mathbb{N}(x,u)-u^{-1}f(x)-u^{-2}g(x)=\sum _{j=1}^n{\mu }_{jx}{D}_{{\theta }_j}^{{\gamma }_j}\mathbb{N}(x,u)+\Phi (x,u)$$

$${}_x{D}_{{\theta }_j}^{{\gamma }_j}f(x)$$

$$F\left[u^{-\alpha }\mathbb{N}(x,u)-u^{-1}f(x)-u^{-2}g(x)\right]$$

$$u^{-\alpha }{\mathbb{N}}^{\ast }(k,u)-u^{-1}{f}^{\ast }(k)-u^{-2}{g}^{\ast }(k)$$

$$F\left[{}_x{D}_{{\theta }_j}^{{\gamma }_j}f(x);k\right]=-{\Psi }_{{\theta }_j}^{{\gamma }_j}(k)f^{\ast }(k)$$

$${\Psi }_{{\theta }_j}^{{\gamma }_j}(k)=|k{|}^{{\gamma }_j}\exp [i\text{sign}(k)\frac{{\theta }_j\pi }{2}],0<{\gamma }_j\le 2,|{\theta }_j|\le min\{{\gamma }_j,2-{\gamma }_j\}$$

$${\mathbb{N}}^{\ast }(k,u)\left[u^{-\alpha }+\sum _{j=1}^n{\mu }_j{\Psi }_{{\theta }_j}^{{\gamma }_j}(k)\right]=u^{-1}{f}^{\ast }(k)+u^{-2}{g}^{\ast }(k)+{\Phi }^{\ast }(k,u)$$

$${\mathbb{N}}^{\ast }(k,u)=\frac{u^{-1}{f}^{\ast }(k)}{u^{-\alpha }+b(k)}+\frac{u^{-2}{g}^{\ast }(k)}{u^{-\alpha }+b(k)}+\frac{{\Phi }^{\ast }(k,u)}{u^{-\alpha }+b(k)}$$

$$S^{-1}[{\mathbb{N}}^{\ast }(k,u)]$$

Full text

**Key words   :**Laplace Transform, Sumudu transform, Reaction-Diffusion Systems, Mittag-Leffler function

**2010 Mathematics Subject Classification   :**44A10, 44A20, 35K57, 33E12

1. Introduction

In recent decades, researchers care study and applications of fractional calculus, which is considered one of the most interesting topics in applied mathematics. The applications of fractional integral operator involving various special functions has found in various sub fields such as statistical distribution theory, control theory, fluid dynamics, stochastic dynamical system, plasma, image processing, nonlinear biological systems, astrophysics, and in quantum mechanics (see [2], [13], [3]). The great use of different transforms in quantum mechanics gives wide variety of results, so it attracted mathematicians and physicists to pay more attention ([14]-[16]). Many more related works found in ([18], [21], [22], [23], [24],[10],[11],[8],] and there in.

2. Mathematical background

Starting this section, by providing the definition of Laplace transform and its inverse, then the definition of Sumudu transform and its inverse for the function of function $f(x,t)$. these definitions are given as:

If a continuous or piecewise continuous function $f(x,t)$ is assumed as a function of order ($e^{t}$) when $t\rightarrow\infty$, then the Laplace transform with respect to $t$ and its inverse with respect to $s$ is respectively given by

[TABLE]

such that, $\Re(s)>0,$ and

[TABLE]

where $\nu\in\mathbb{R}$ is fixed.

In this paper, we concern our attention to two definitions based on Riemann-Liouville (R-L) definition of fractional integral of order $\nu>0$ which given by [17],

[TABLE]

• •

) The first one is Riemann-Liouville definition of fractional derivative of order $\alpha>0$, and it may be given as follows [20]

[TABLE]

where $[\alpha]$ is the integer part of the number $\alpha$.

• •

) The second is Caputo derivative [6], given in the form

[TABLE]

where $f^{(m)}(x,t)=\dfrac{\partial^{m}f}{\partial t^{m}}$ is m${}^{th}~{}$.

The Laplace transform of both definitions[19]:

[TABLE]

where $m-1<\alpha\leq m$, and $\Re(s)>0.$

The Sumudu transform over the set functions

[TABLE]

is defined by

[TABLE]

The detailed literature of Sumudu transform is found in ([1],[4],[5]). The following results due to [26]

[TABLE]

by interpreting it with the help of the formula $S^{-1}\left[u^{n}:t\right]=\frac{t^{n}}{\Gamma(n+1)}$ ,gives

[TABLE]

and for $\lambda=0$ we get

[TABLE]

where $E_{\alpha,\beta}^{\delta}(z)$ is the Mittag-Leffler function [9] is in the form

[TABLE]

where $(\delta)_{n}$ is Pochhammer symbol.The Sumudu convolution theorem in [5] is

[TABLE]

The Sumudu transform of the Riemann-Liouville derivative [4] is given as:

[TABLE]

and for Caputo’s derivative is given as:

[TABLE]

3. Main Results

Here we deals with the solution of fractional of reaction diffusion equation

Theorem 1**.**

The one dimensional fractional non-homogeneous reaction diffusion system is defined by

[TABLE]

where $t>0,\,x\in\mathbb{R},$ and, the parameters $\alpha,\,\theta_{1},...,\theta_{n},\,\gamma_{1},...,\gamma_{n}$ are real. These parameters satisfy the following condition:

[TABLE]

with initial conditions

[TABLE]

where ${}_{0}D_{t}^{\nu}N(x,t)$ is the R-L fractional derivative of N$(x,t)$ with respect to t of order $\nu$. The Riesz-Feller (R-L) space fractional derivative ${}_{x}D_{\theta_{1}}^{\gamma_{1}},...,_{x}D_{\theta_{n}}^{\gamma_{n}}$ with asymmetries $\theta_{1},...,\theta_{n}~{}$ respectively,$\mu_{1,...,}\mu_{n}$ are arbitrary constants, and, the functions $f(x),g(x)$ and $\phi(x,t)$ are given. Then the solution of $(\ref{7a})$ is:

[TABLE]

where $E_{\alpha,\beta}(.)$ is defined in $(\ref{Mittag})$

Proof.

By using Sumudu transform with the variable $t$ and applying the given conditions $(\ref{8})$ and $(\ref{9})$ we get

[TABLE]

[TABLE]

[TABLE]

where $S[\phi(x,t);u]=\Phi(x,u)$ denotes Sumudu transform. The integral representation of (R-L) in $x$ domain [25] is

[TABLE]

By using Fourier-Transform with the variable x, we get

[TABLE]

where * denoted by the Fourier transform. For R-L fractional derivative the Fourier transform is

[TABLE]

such that

[TABLE]

Solving $\mathbb{N}^{\ast}(k,u)$ yields,

[TABLE]

[TABLE]

where $b(k)=\sum_{j=1}^{n}\mu_{j~{}}\Psi_{\theta_{j}}^{\gamma_{j}}(k)$

Applying the the inverse Sumudu transform and using the convolution theorem for the last term $(\ref{5})$

[TABLE]

and the result given in ,we get

[TABLE]

Finally using the inverse Fourier transform, we get the required result. ∎

3.1. Particular cases:

If we take $\theta_{1}=\theta_{2}=...=\theta_{n}=0$ then, instead of $b(k)$ we get

[TABLE]

such that $C$ is constant given by $C=\sum_{j=1}^{n}\mu_{j}$ and we have the following corollaries.

Corollary 3.1**.**

Consider

[TABLE]

which is the one dimensional non homogeneous unified fractional reaction diffusion model associated with time derivative ${}_{0}D_{t}^{\alpha}$ and Riez space derivatives ${}_{x}D_{0}^{\gamma_{1}},...,_{x}D_{0}^{\gamma n}$, $n\in\mathbb{N}$ ,t$>0,x\in R;\alpha,\gamma_{1},...,\gamma_{n}$ are real parameters with constraints

[TABLE]

with initial condition

[TABLE]

then the following formula hold true

[TABLE]

Corollary 3.2**.**

If consider $g(x)=0$ under the condition

[TABLE]

with initial condition

[TABLE]

then the following formula hold true

[TABLE]

Theorem 2**.**

Consider the following fractional order unified one-dimensional non-homogeneous reaction diffusion equation

[TABLE]

where $t>0,\lambda,x\in\mathbb{R},~{}\alpha,\beta,\theta_{1},...,\theta_{n},\gamma_{1},...,\gamma_{n}$ are real parameters with the conditions:

[TABLE]

with initial conditions

[TABLE]

then the following formula holds true:

[TABLE]

Proof.

Applying the Sumudu transform with respect to time variable $t$ and using $(\ref{19})$ and $(\ref{20})$ we get

[TABLE]

[TABLE]

[TABLE]

Applying the Fourier-Transform with respect to the space variable $x$ , we get

[TABLE]

Solving $\mathbb{N}^{\ast}(k,u)$ yields,

[TABLE]

[TABLE]

where $b(k)=\sum_{j=1}^{n}\mu_{j~{}}\Psi_{\theta_{j}}^{\gamma_{j}}(k)$ Applying the the inverse Sumudu transform and using the convolution theorem for the last term $(\ref{5})$

[TABLE]

we get

[TABLE]

Finally applying the inverse Fourier transform, we get

[TABLE]

∎

To get the results of theorem (1) we expand the summation into two parts $r=0$ and $\sum_{r=1}^{\infty}$, then we substitute by $\lambda=0$