Skip to main content

Theory and Modern Applications

On some applications of the space-time fractional derivative


In this work, we study and extend the one-dimensional fractional derivative to the multidimensional space-time fractional derivative, determine the exact solution via the Laplace transform, and develop mathematical foundations of the respective operators. The multidimensional space-time fractional derivative is developed to augment the equations. The results show that this method is effective, convenient, and easily executable. The main advantage of the proposed approach is that it is an accurate analytical method (the Laplace transform method) that can be implemented for both space and time discretizations of the fractional derivatives and allows us to present new solutions to problems by certain applications for solving space-time fractional derivatives.

1 Introduction

The integrals of integer order have clear physical interpretations in engineering and geometry and are features that help to solve applied problems in various scientific fields. Fractional calculus describes and is used to model physical processes via fractional differential equations. In the past, the investigation of nonlinear equations of traveling-wave solutions played an important role in the study of nonlinear physical phenomena (see [14]). Fractional differential equations are used to model various physical phenomena in porous, colloid materials, disordered, random, amorphous, geology, finance, and medicine [5, 6]. In [1, 3, 7, 8], many new mathematical models were developed to represent the limiting distribution of a specified stochastic, and the developed models successfully employ fractional derivatives in respective diffusion equations. Also, many physical situations were observed using anomalous diffusion.

Bhrawy et al. [9] proposed the implementation of a spectral method for time and space discretizations. They presented a new and efficient algorithm for solving a time fractional subdiffusion equation on a semiinfinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization, along with the Laguerre-Gauss-Radau collocation scheme, which is used for space discretization on a semiinfinite domain.

Bhrawy [10] adapted an operational matrix formulation of the collocation method for one- and two-dimensional nonlinear fractional subdiffusion equations. They also used both double and triple shifted Jacobi polynomials as basis functions to elucidate approximate solutions of one- and two-dimensional cases. The space-time fractional derivatives given in the underlined problems are expressed by Jacobi operational matrices. This helps to investigate spectral collocation schemes for both temporal and spatial discretizations. Bhrawy [11] also employed the shifted Legendre Gauss-Lobatto collocation scheme and Chebyshev Gauss-Radau collocation approximations for temporal and spatial discretizations. He then focused on implementing the new algorithm for two physical problems, time fractional modified anomalous subdiffusion, and fractional nonlinear superdiffusion equations.

The main type of spectral methods, collocation [12], shows that spectral methods not only have exponential rates of convergence, but also high levels of accuracy.

Bhrawy et al. [13] reported a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers equation based on spectral shifted Legendre collocation method in combination with the shifted Legendre operational matrix of fractional derivatives in the Caputo sense and proposed this method in both spatial and temporal discretizations for the space-time fractional Burgers equation to reduce such problems to that of solving a system of nonlinear algebraic equations.

Furthermore, Mainardi et al. [14] discussed the fundamental solutions of space-time diffusion equation of fractional order using the Fourier and Laplace integral transforms and Mittag-Leffler functions, in which the fundamental solution can only be expressed as a convolution form of a Green function and the initial value function, which is then computed.

In the case of the fundamental solutions of space and space-time Riesz fractional partial differential equations with regular conditions, in [15], the authors considered the fundamental solutions of the space-time fractional partial differential equations of the Riesz type with regular conditions; they are obtained from the partial differential equation by replacing the time derivative with fractional derivative of Caputo type and the space derivative with Riesz potential. Also, they derived explicit expressions of the fundamental solutions for the space Riesz fractional partial differential equation and the space-time Riesz fractional partial differential equation. Ebadian et al. [16] solved the fractional diffusion-wave equation using algorithms that are dependent on triangular function methods.

Wijker [17] discussed a random vibration of linear dynamic systems and considered spacecraft structures as behaving linearly and having an important role in the design and verification process for characteristic models. Similarly, Brooks et al. [18] constructed functions (spin-wave) of the inversion symmetry, obtained a relation between the basic functions of different members for the wave vector, introduced order parameters, and determined the properties of transformation under the operations of a paramagnetic crystal.

Ahmet [19] utilized the method of tanh-coth with the Riccati equation to handle nonlinear wave equations and to travel wave solutions of nonlinear evolution equations. Mark and Hans [20] developed limit theorems by triangular arrays for sequences of continuous-time random walks processes and solved fractional diffusion equations by transition densities of continuous-time random walks.

Wu et al. [21] proposed a Riesz Riemann-Liouville difference for modeling the diffusion equation on discrete media, proposed a lattice fractional diffusion equation, reduced Burgers equations to a system of differential equations with initial conditions, and adopted the Adomian decomposition method to obtain numerical solutions for the fractional partial difference equations.

Wu et al. [22] also proposed a fractional logistic map and Lorenz maps of Riemann-Liouville type and designed chaos synchronization of fractional chaotic maps according to the stability results. They extended the control method to discrete fractional equations and selected the control parameters according to the stability conditions.

In [23], Wu et al. proposed a Riesz-Caputo fractional difference within the discrete fractional calculus, suggested a fractional difference equation method for anomalous diffusion in discrete finite domains, and discussed the numerical simulation of a lattice fractional diffusion process for various difference orders.

Wu and Baleanu [24] studied master-slave synchronization for the fractional difference equation with a nonlinear coupling method. The designed synchronization method numerically can effectively synchronize the fractional logistic map. They adopted the Caputo-like delta derivative as the difference operator.

2 Space-time fractional derivative

Let v be a probability distribution on \(\Re_{+}\times\Re^{d}=[0,\infty ]\), where \(\Re_{+}=[0,\infty]\). Define the Fourier-Laplace transform as

$$ \hat{v}= \int_{\Re^{d}} \int_{\Re_{+}}e^{ix\cdot k}e^{-st}v(dt,dx), $$

where \(v^{n}=v * \cdots * v\) is the n-fold convolution of v, v is infinitely divisible for \(n=1,2,\ldots\) , and \(v_{n}\) is a probability distribution such that \(v^{n}_{n}=v\).

Theorem 1

Let v be finitely divisible on \(\Re_{+}\times\Re^{d}\) and define

$$ T(x)f(y,t)= \int^{t}_{0} \int_{\Re_{d}}f(y-s_{1},t-s_{2})v^{x}(ds_{1},ds_{2}) $$

for \(f\in L^{1}_{w} \{\Re_{+}\times\Re^{d} \}\) and \(x\geq0\). Then \({M(x)}_{x\geq0}\) has the following properties for all \(f\in L^{1}_{w}(\Re_{+}\times\Re^{d})\):

  1. (i)

    \(M(x+v)f=M(x)M(v)\) for \(v,x\geq0\).

  2. (ii)


  3. (iii)

    \(f>0\Rightarrow M(x)f>0\) for \(x>0\).

  4. (iv)

    \(\Vert M(x)f\Vert _{w}\leq \Vert f\Vert _{w}\) for all \(x\geq0\).

  5. (v)

    \({\lim_{x\rightarrow0}\Vert M(x)f-f\Vert _{w}=0}\).


Let \(f=I_{P}(y,t)=I\) (\((y,t)\in P\)), \(P=(y,t)\in\Re_{+}\times\Re^{d}:a_{0}\leq t\leq b_{0}\), and \(a_{i}\leq y_{i}\leq b_{i}\), \(i=1,2,\ldots,d\).


$$\begin{aligned} \int^{\infty}_{0} \int_{\Re^{d}}\bigl\vert M(x)f(y,t)\bigr\vert \,dy\,dt =& \int ^{\infty}_{0} \int_{\Re_{d}}\biggl\vert \int^{t}_{0} \int_{\Re _{d}}f(y-s_{1},t-s_{2})v^{u}(ds_{1},ds_{2}) \biggr\vert \,dy\,dt \\ =& \int^{\infty}_{0} \int_{\Re_{d}} \int \int _{s_{1},s_{2}+P}ds_{1}\, ds_{2}v^{y}(ds_{1},ds_{2}), \end{aligned}$$

where v is infinitely divisible, and we have \(v^{x}\rightarrow v^{0}\) as \(u\downarrow0\) and \(v^{x}(B)\rightarrow v(B)\) for all Borel subsets \(B\subseteq\Re_{+}\times\Re^{d}\) such that \(v^{0}(\partial B)=0\), where \(\partial B=\overline{B}\setminus B^{0}\), is the closure of B, that is, the intersection of all closed sets containing B, and \(B^{0}\) is the interior, that is, the union of all open sets contained in B.

If \((0,0)\in\partial B\), then, for all \((y,t)\in\Re_{+}\times\Re^{d}\) with \(t\neq a_{0},b_{0}\) and \(y_{i}\neq a_{i},b_{i}\) for all \(i=1,2,\ldots,d\), we can get

$$\lim_{x\rightarrow0^{+}} M(x)f(y,t)=\lim_{x\rightarrow 0^{+}}v^{x} \bigl((y,t)-P\bigr)=v^{0}\bigl((y,t)-P\bigr)=f(s_{2},t). $$

By the dominated convergence theorem,

$$\begin{aligned} \int \int_{P}\bigl\vert M(x)f(y,t)-f(y,t)\bigr\vert \,dy \,dt =& \int \int_{P} \bigl\vert v^{x}\bigl((y,t)-P \bigr)-v^{0}\bigl((y,t)-P\bigr)\bigr\vert \,dy\,dt \\ =& \int^{\infty}_{0} \int_{\Re_{d}} \int \int _{s_{1},s_{2}+P}dy\, dtv^{x}(ds_{1},ds_{2}) \end{aligned}$$

implies that

$$ \int \int_{(y,t)\notin P}\bigl\vert M(x)f(y,t)-f(y,t)\bigr\vert \,dy\,dt = \int \int _{(y,t)\notin P}\bigl\vert M(x)f(y,t)\bigr\vert \,dy\,dt \rightarrow0. $$


$$ \int \int_{(y,t)\notin P}\bigl\vert M(x)f(y,t)-f(y,t)\bigr\vert \,dy\,dt \rightarrow0. $$

By the triangle inequality,

$$ \bigl\Vert M(x)f-f\bigr\Vert _{w}\leq\bigl\Vert M(x)f-T(x)g^{w}_{n}\bigr\Vert _{w}+\bigl\Vert M(x)g^{w}_{n}-g^{w}_{n}\bigr\Vert _{w}+\bigl\Vert g^{w}_{n}-f \bigr\Vert _{w}. $$

By property (iv),

$$ \bigl\Vert M(x)f-M(x)g^{w}_{n}\bigr\Vert _{w}\leq\bigl\Vert f-g^{w}_{n}\bigr\Vert _{w}\leq n^{-1}. $$

Thus, we have

$$ \bigl\Vert M(x)f-f\bigr\Vert _{w}\leq n^{-1}+\bigl\Vert M(x)g^{w}_{n}-g^{w}_{n}\bigr\Vert _{w}+n^{-1}. $$

For a continuous semigroup \(M(x)\), \(x>0\), on Y, the generator is given by

$$ Lf=\lim_{x\rightarrow0}\frac{M(x)f-f}{x}. $$

Then \(\Vert x^{-1}(M(x)f-f)-Lf\Vert \rightarrow0\) in \(\Vert Y \Vert \). □

Theorem 2

If \(X=L^{1}_{w}=\Re^{+}\times\Re_{c}\), then L is the generator of a continuous semigroup, and \(M(u)\) is defined as

$$ M(u)f= \int^{t}_{0} \int_{\Re_{+}}f(x-y,t-s)v^{u}(dy,ds). $$


$$ \widehat{Lf}(k,s):=\psi(k,s)\hat{f}(k,s), $$

where ψ is given, \(H(L)= \{f\in X:\psi\hat{f}=\hat {h}(k,s)\ \exists h\in X \}\), and Σ is a subset of \(L^{1}_{w}=\Re^{+}\times\Re_{c}\). Then \(\Sigma\subset H(L)\) for \(f\in\Sigma \), and we can get

$$\begin{aligned} Lf(x,t) =& -a\cdot\nabla f(x,t)+b\cdot\frac{\partial f}{\partial t}(x,t)+\frac {1}{2} \nabla\cdot A\nabla f(x,t) \\ &{} + \int_{\Re^{d}\times\Re_{+} (0,0)}\biggl(G(t-s)f(x-y,t-s)-f(x,t)+\frac {\nabla f(x,t)\cdot y}{1-\Vert y\Vert ^{2}} \biggr)\phi(dy,ds), \end{aligned}$$

where \(G(t)\) is the Heaviside step function.


If \(f\in H(L)\subset L^{1}_{w}(\Re^{+}\times\Re_{c})\), then its Fourier-Laplace transform exists. The Fourier-Laplace transform of a convolution is \(\widehat{T(u)f}(k,s)=\exp(u\psi(k,s))\hat{f}(k,s)\). Then

$$\begin{aligned} \widehat{L \{f \}}(k,s) =&\lim_{u\rightarrow0^{+}}\frac {\exp(u\psi(k,s))-1}{u}\cdot \hat{f}(k,s) \\ =&\psi(k,s)\hat{f}(k,s) \end{aligned}$$

for \(f\in H(L)\).

Let \(f\in X\). Then \(g:=\lambda f-h\in X\). By semigroup theory, \((\lambda I-L)^{-1}\) is a bounded linear for \(\lambda>0\).

Now, let \(q=(\lambda I-L)^{-1}g\). Then

$$\hat{q}(k,s)=\frac{\hat{g}(k,s)}{\lambda-\psi(k,s)}= \frac{\lambda\hat{f}(k,s)-\psi(k,s)\hat{f}(k,s)}{\lambda-\psi (k,s)}=\hat{f}(k,s). $$


$$\Vert f\Vert _{\Sigma}=\Vert f\Vert _{w}+\sum ^{c}_{i=1}\biggl\Vert \frac {\partial f}{\partial x_{i}}\biggr\Vert _{w}+\sum^{c}_{i,j=1} \biggl\Vert \frac {\partial^{2}f}{\partial x_{i}\partial x_{j}}\biggr\Vert _{w}+\biggl\Vert \frac {\partial f}{\partial t}\biggr\Vert _{w}, $$

and using a Taylor series expansion on x, we have

$$ f(x-y,t)-f(x,t)+y\cdot\nabla f(x,t)=y\cdot \int^{1}_{0}(1-r)M_{x-ry}y\, dr, $$

where \(P_{x}\) is the Hessian matrix of f, and

$$ \int e^{-wt}\biggl\vert f(x-y,t)-f(x,t)+\frac{y\cdot\nabla f(x,t)}{1+\Vert y\Vert ^{2}} \biggr\vert \,dx\,dt\leq C\Vert f\Vert _{\Sigma}\frac{\Vert y\Vert ^{2}}{1+\Vert y\Vert ^{2}} $$

for some C. Further, by Taylor series expansion we obtain

$$ \int e^{-wt}\bigl\vert f(x,t-s)-f(x,t)\bigr\vert \,dx \,dt \leq H\Vert f\Vert \cdot\frac{s}{1+s} $$

for some constant H.

Then, by Fubini’s theorem we have

$$\begin{aligned}& \int e^{-wt}\biggl( \int\biggl\vert f(x-y,t-s)-f(x,t)+\frac{y\cdot\nabla f(x,t)}{1+\Vert y\Vert ^{2}}\biggr\vert \phi(dy,ds)\biggr)\,dx\,dt \\& \quad = \int\biggl( \int e^{-wt}\biggl\vert f(x-y,t-s)-f(x,t)+\frac{y\cdot\nabla f(x,t)}{1+\Vert y\Vert ^{2}} \biggr\vert \,dx\,dt\biggr)\phi(dy,ds) \\& \quad \leq \Vert f\Vert _{\Sigma}\biggl(C \int\frac{\Vert y\Vert ^{2}}{1+ \Vert y\Vert ^{2}}\phi(dy,ds) \\& \qquad +H \int\frac{s}{1+s}\phi(dy,ds)\biggr)\leq K\Vert f\Vert _{\Sigma}. \end{aligned}$$

We show that \(\Sigma\subset H(L)\). If

$$ L_{1}f(x,t)=-a\cdot\nabla f(x,t)+b\cdot\frac{\partial f}{\partial t}f(x,t)+ \frac {1}{2}\nabla\cdot A\nabla\cdot f(x,t), $$


$$ \Vert L_{1}f\Vert _{w}\leq B\Vert f\Vert _{\Sigma} $$

for B independent of f,

$$ L_{2}f(x,t)= \int_{\Re^{+}\times\Re_{c}\setminus(0)} \biggl(f(x-y,t-s)-f(x,t)+\frac{\nabla f(x,t)\cdot y}{1-\Vert y\Vert ^{2}} \biggr) \phi(dy,ds). $$

Then, \(L=L_{1}-L_{2}\) and

$$ f\in\Sigma\cap C^{\infty}\bigl(\Re^{c}\times\Re_{+} \bigr), \quad \Vert Lf\Vert _{w} \leq \Vert L_{1}f \Vert _{w}+ \Vert L_{2}f\Vert _{w}\leq B \Vert f\Vert _{\Sigma}+K\Vert f\Vert _{\Sigma}. $$

If f is denoted by

$$\{f_{n}\}^{\infty}_{n=1}\subset\Sigma\cap C^{\infty}\bigl(\Re^{+}\times\Re_{c}\bigr), $$

then \(f_{n}\rightarrow f\) in the \(\Vert L^{1}_{w}\Vert \), and the above equation shows that \(\{f_{n}\}^{\infty}_{n=1}\) converges to g in the \(f_{n}\rightarrow f\). Since \(\{L \}\) is closed, \(f\in D(L)\) and \(L \{f \}=g\). □

3 Solution of the linear space-time fractional differential equations

Consider the space-time fractional order for a one-dimensional differential equation. The general form of derivative with respect to t is given by

$$ \sum^{m}_{i=1}\frac{\partial^{v_{i}}u(x_{i},t_{i})}{\partial t^{v_{i}}_{i}} = \sum ^{m}_{i=1}\frac{\partial ^{n_{i}}u(x_{i},t_{i})}{\partial t^{n_{i}}_{i}} \Biggl[\sum ^{m}_{i=1}\frac {\partial^{-n_{i}+v_{i}}u(x_{i},t_{i})}{\partial t^{-n_{i}+v_{i}}_{i}} \Biggr], $$

where \(n_{i}\) is the smallest integer greater than \(v_{i}>0\) and \(u_{i}=n_{i}-v_{i}\) for each \(i=1,2,\ldots,m\). We can write this equation as

$$ \sum^{m}_{i=1}\frac{\partial^{v_{i}}u(x_{i},t_{i})}{\partial t^{v_{i}}_{i}} = \sum ^{m}_{i=1}\frac{\partial ^{n_{i}}u(x_{i},t_{i})}{\partial t^{n_{i}}_{i}} \Biggl[\sum ^{m}_{i=1}\frac {\partial^{-(n_{i}-v_{i})}u(x_{i},t_{i})}{\partial t^{-(n_{i}-v_{i})}_{i}} \Biggr] $$

together with the initial condition. By the Laplace transform we can get

$$\begin{aligned} L \Biggl\{ \sum^{m}_{i=1} \frac{\partial^{v_{i}}u(x_{i},t_{i})}{\partial t^{v_{i}}_{i}} \Biggr\} =&L \Biggl\{ \sum^{m}_{i=1} \frac{\partial ^{n_{i}}u(x_{i},t_{i})}{\partial t^{n_{i}}_{i}} \Biggl[\sum^{m}_{i=1} \frac {\partial^{-(n_{i}-v_{i})}u(x_{i},t_{i})}{\partial t^{-(n_{i}-v_{i})}_{i}} \Biggr] \Biggr\} \\ =&\sum^{m}_{i=1} \Biggl[s^{n_{i}}_{i}L \Biggl\{ \sum^{m}_{i=1}\frac{\partial ^{-(n_{i}-v_{i})}u(x_{i},t_{i})}{\partial t^{-(n_{i}-v_{i})}_{i}} \Biggr\} -\sum^{m}_{i=1}s^{n_{i}-i}_{i} \frac{\partial ^{i-1-n_{i}+v_{i}}u(x_{i},0)}{\partial t^{i-1-n_{i}+v_{i}}_{i}} \Biggr] \\ =&\sum^{m}_{i=1} \Biggl[s^{n_{i}}_{i} \bigl(s^{-n_{i}+v_{i}}_{i}U(x_{i},s_{i}) \bigr)-\sum^{m}_{i=1}s^{n_{i}-i}_{i} \frac{\partial ^{i-1-n_{i}+v_{i}}u(x_{i},0)}{\partial t^{i-1-n_{i}+v_{i}}_{i}} \Biggr] \\ =&\sum^{m}_{i=1} \Biggl[s^{v_{i}}_{i}U(x_{i},s_{i})- \sum^{m}_{i=1}s^{n_{i}-i}_{i} \frac{\partial ^{i-1-n_{i}+v_{i}}u(x_{i},0)}{\partial t^{i-1-n_{i}+v_{i}}_{i}} \Biggr]. \end{aligned}$$

Now, if \(n=1\), then the transform is given by

$$\sum^{m}_{i=1} \biggl[s^{v_{i}}_{i}U(x_{i},s_{i})- \frac{\partial ^{-1+v_{i}}u(x_{i},0)}{\partial t^{-1+v_{i}}_{i}} \biggr], \quad 0< v_{i}\leq 1. $$

If \(n=2\), then

$$\Rightarrow\quad \sum^{m}_{i=1} \biggl[s^{v_{i}}_{i}U(x_{i},s_{i})-s_{i} \frac {\partial^{-2+v_{i}}u(x_{i},0)}{\partial t^{-2+v_{i}}_{i}}-\frac {\partial^{-1+v_{i}}u(x_{i},0)}{\partial t^{-1+v_{i}}_{i}} \biggr],\quad 1< v_{i}\leq2. $$

If \(n=3\), then

$$\sum^{m}_{i=1} \biggl[s^{v_{i}}_{i}U(x_{i},s_{i})-s^{2}_{i} \frac{\partial ^{-3+v_{i}}u(x_{i},0)}{\partial t^{-3+v_{i}}_{i}}-s_{i}\frac{\partial ^{-2+v_{i}}u(x_{i},0)}{\partial t^{-2+v_{i}}_{i}}-\frac{\partial ^{-1+v_{i}}u(x_{i},0)}{\partial t^{-1+v_{i}}_{i}} \biggr], $$

where \(2< v_{i}\leq3\).

Now, if \(n=4\), then we obtain

$$\begin{aligned}& \sum^{m}_{i=1} \biggl[s^{v_{i}}_{i}U(x_{i},s_{i})-s^{3}_{i} \frac{\partial ^{-4+v_{i}}u(x_{i},0)}{\partial t^{-4+v_{i}}_{i}}-s^{2}_{i}\frac {\partial^{-3+v_{i}}u(x_{i},0)}{\partial t^{-3+v_{i}}_{i}} \\& \quad {}-s_{i} \frac {\partial^{-2+v_{i}}u(x_{i},0)}{\partial t^{-2+v_{i}}_{i}}-\frac {\partial^{-1+v_{i}}u(x_{i},0)}{\partial t^{-1+v_{i}}_{i}} \biggr], \end{aligned}$$

where \(3< v_{i}\leq4\).

Similarly, the general form of the derivative with respect to x is given by

$$ L \Biggl\{ \sum^{m}_{i=1} \frac{\partial^{v_{i}}u(x_{i},t_{i})}{\partial x^{v_{i}}_{i}} \Biggr\} = \sum^{m}_{i=1} \Biggl[s^{v_{i}}_{i}U(s_{i},t_{i})- \sum^{m}_{i=1}s^{n_{i}-i}_{i} \frac{\partial ^{i-1-n_{i}+v_{i}}u(0,t_{i})}{\partial x^{i-1-n_{i}+v_{i}}_{i}} \Biggr]. $$

Now, we provide some basic definitions and examples where the method is applied for solving linear space-time fractional differential equations. In fact, many different disciplines, such as biology, finance, semiconductor research and hydrology (see [25]), and space fractional differential equation (see [26, 27]) all utilize irregular transport in the context of the flow porus media, fractional space derivatives model, and huge motions through largely conjunctive fractures. Now, we recall a method to solve the space-time fractional differential equations.

Definition 1

see [28]

The Laplace transform of \(f(x)\) for \(x>0\) is denoted by \(L \{ f(x) \}\) or \(F(s)\) and defined by

$$ L \bigl\{ f(x) \bigr\} =F(s)= \int^{\infty}_{0}e^{-sx}f(x)\,dx, $$

where s is a complex number.

Example 1


$$\frac{\partial^{\frac{3}{2}}}{\partial x^{\frac{3}{2}}}y(x,t)-\frac {\partial^{\frac{1}{2}}}{\partial x^{\frac{1}{2}}}y(x,t)=e^{-\frac {3}{2}x} \biggl[\cos ax+ \biggl(\frac{\frac{1}{2}+\frac{3}{2}}{1} \biggr)\sin ax \biggr], $$

where \(a=1\), \(1<3/2\leq2\), and \(0<1/2\leq1\).

Solution: By using the Laplace transform to the above equation we get:

$$\begin{aligned}& L \biggl\{ \frac{\partial^{\frac{3}{2}}}{\partial x^{\frac {3}{2}}}y(x,t)-\frac{\partial^{\frac{1}{2}}}{\partial x^{\frac {1}{2}}}y(x,t) \biggr\} \\& \quad =L \biggl\{ e^{-\frac{3}{2}x}\biggl[\cos x+\biggl(\frac{\frac {1}{2}+\frac{3}{2}}{1}\biggr)\sin x\biggr] \biggr\} , \\& s^{\frac{3}{2}}Y(s,t)+s^{\frac{1}{2}}Y(s,t) = \frac{s+\frac {1}{2}}{(s+\frac{3}{2})^{2}+1}, \\& Y(s,t) = \frac{s+\frac{3}{2}}{s^{\frac{1}{2}}(s+1)[(s+\frac {3}{2})^{2}+1]} \\& \hphantom{Y(s,t)} = \frac{s+\frac{1}{2}}{s^{\frac{1}{2}}(s+1)[(s+\frac {3}{2})^{2}+1]}+\frac{1}{s^{\frac{1}{2}}(s+1)[(s+\frac{3}{2})^{2}+1]} \\& \hphantom{Y(s,t)} = \frac{1}{s^{\frac{1}{2}}(s+1)} \biggl[\frac{s+\frac{1}{2}}{(s+\frac {3}{2})^{2}+1}+\frac{1}{(s+\frac{3}{2})^{2}+1} \biggr]. \end{aligned}$$

By using the inverse three-dimensional Laplace transform to the above equation we can get:

$$ y(x,t)=x^{\frac{3}{2}}E_{1,\frac{3}{2}+1}(x) \bigl[e^{-\frac {3}{2}x}[\cos x+\sin x]+e^{-\frac{3}{2}x}\sin x \bigr]. $$

Example 2

(i) Letting \(\gamma=\frac{4}{3}\), solve \(y''(x,t)-y^{\gamma }(x,t)=\delta'(x)\).

Solution: By taking the Laplace transform of the above equation we can get:

$$ L \bigl\{ y'' \bigr\} -L \bigl\{ y^{\gamma} \bigr\} =L \bigl\{ \delta '(x) \bigr\} , $$

where \(\gamma=\frac{4}{3}\),

$$ s^{2}Y(s,t)-sy(0,t)-y'(0,t)-s^{\frac{4}{3}}Y(s,t)+s D^{-(2-4/3)}y(0,t)+D^{-(1-4/3)}y(0,t)=s $$

together with the initial conditions

$$ y(0,t)=y'(0,t)=D^{-(2/3)}y(0,t)+D^{(1/3)}y(0,t)=0. $$

We can get

$$ s^{2}Y(s,t)-s^{\frac{4}{3}}Y(s,t)=s. $$

Now, by solving for \(Y(s)\), we obtain:

$$ Y(s,t) = \frac{s}{s^{2}-s^{\frac{4}{3}}}. $$


$$\begin{aligned} y(x,t) =&L^{-1} \biggl\{ \frac{1}{s^{1/3}(s^{\frac{2}{3}}-1)} \biggr\} \\ =&L^{-1} \biggl\{ \frac{1}{s^{1/3}} \biggr\} L^{-1} \biggl\{ \frac{1}{s^{\frac {2}{3}}-1} \biggr\} \\ =& \biggl(\frac{x^{\frac{1}{3}-1}}{\Gamma(\frac{1}{3})} \biggr) \bigl(x^{(2/3)-1}E_{(2/3),(2/3)}x^{(2/3)} \bigr). \end{aligned}$$

(ii) Now let \(\gamma=\frac{8}{3}\) and \(1<\alpha=\frac{5}{3}\leq 2\). Then solve \(y^{\gamma}(x,t)+y'(x,t)=(\delta'(x)-\delta (x))(-x^{\alpha} \delta_{1,\frac{-2}{3}}(x))\), \(2<\gamma\leq3\).

Solution: By taking the Laplace transform of the above equation we can get:-

$$ L \bigl\{ y^{\gamma} \bigr\} +L \bigl\{ y' \bigr\} =L \bigl\{ \bigl(\delta '(x)-\delta(x)\bigr) \bigl(x^{\alpha} \delta_{1,\frac{-2}{3}}(x)\bigr) \bigr\} , $$

where \(\gamma=\frac{8}{3}\) and \(\alpha=\frac{5}{3}\).

Together with the initial conditions, we can get:

$$ s^{\frac{8}{3}}Y(s,t)+sY(s,t)=(s-1) \biggl(\frac{s^{\frac {5}{3}}}{s-1} \biggr). $$

Now, by solving for \(Y(s,t)\) we obtain:

$$ Y(s,t) = \frac{s^{\frac{5}{3}}}{s(s^{5/3}+1)}. $$


$$ y(x,t) = \delta_{\frac{5}{3}}\bigl(-x^{\frac{5}{3}}\bigr). $$

Note that, in particular, if \(\gamma=1, 2\), then the derivative is the classical ordinary derivative.

4 Solution of space-time fractional linear partial differential equation

Consider the initial and boundary value problem of the space time fractional linear partial differential equation: The general form for multivariables with derivative with respect to one variable is given by

$$\begin{aligned} \begin{aligned} &\sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{ij}}_{i}} \\ &\quad = \sum^{\infty}_{j=1}\frac{\partial ^{\sum^{n}_{i=1}m_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{m_{i}}_{i}} \\ &\qquad {} \times \Biggl[\sum^{\infty}_{j=1} \frac{\partial^{\sum ^{n}_{i=1}-(m_{i}-v_{i})}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-v_{i})}_{i}} \Biggr]. \end{aligned} \end{aligned}$$

Together with appropriate initial and boundary conditions, by taking the multidimensional Laplace transform, we can get:

$$\begin{aligned}& L_{n} \Biggl\{ \sum^{\infty}_{j=1} \frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{ij}}_{i}} \Biggr\} \\& \quad = L_{n} \Biggl\{ \sum^{\infty }_{j=1} \frac{\partial^{\sum ^{n}_{i=1}m_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{m_{i}}_{i}} \Biggr\} \\& \qquad {}\times \Biggl[\sum^{\infty}_{j=1} \frac{\partial^{\sum ^{n}_{i=1}-(m_{i}-v_{i})}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-v_{i})}_{i}} \Biggr]. \end{aligned}$$


$$\begin{aligned}& L_{n} \biggl\{ \frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{i}}_{i}} \biggr\} \\& \quad = \prod^{n}_{i=1}s^{m_{i}}_{i}L_{n} \biggl\{ \frac{\partial^{\sum ^{n}_{i=1}-(m_{i}-v_{i})}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-v_{i})}_{i}} \biggr\} \\& {\qquad {}+ \sum^{n-1}_{k=1}(1)^{n-k} \sum^{n}_{\substack{{i_{1},i_{2},\ldots,i_{k}=1}\\{i_{1},i_{2},\ldots,i_{n}}}} \prod ^{k}_{j=1}s^{m_{ij}}_{ij}\prod ^{n}_{\substack{i=1\\ i\neq i_{j}\\ \text{for each }j=1,2,\ldots,k}}\sum^{m_{i}}_{l_{1}=1}s^{l_{1}-1}_{i}} \\& {\qquad {}\times L_{k} \Biggl\{ \frac{\partial^{\sum^{n}_{\substack{{i=1}\\{i\neq i_{j}}}}(m_{i}-l_{i}+v_{i})}u (0,t_{j1},0,t_{j2}, \ldots,0,t_{jn},x_{i1},t_{ij1},0, \ldots,0,x_{i2},t_{ij2},0,\ldots,0,x_{ik},t_{ijk},0, \ldots,0 )}{\prod^{n}_{\substack{{i=1}\\{i\neq i_{j}}}}(m_{i}-l_{i}+v_{i})} \Biggr\} } \\& \qquad {}+ (-1)^{n}\prod^{n}_{i=1} \sum^{m_{i}}_{l_{i}=1}s^{l_{i}-1}_{i} \frac{\partial^{-\sum ^{n}_{i=1}(m_{i}-l_{i}+v_{i})}u(0,t_{j1},0,t_{j2},\ldots,0,t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-l_{i}+v_{i})}_{i}}. \end{aligned}$$

Example 3

Consider the initial and boundary value problem of the space time fractional order linear three-dimensional differential equation

$$ \frac{\partial^{\frac {7}{3}}u(x_{1},t_{1},x_{2},t_{2},x_{3},t_{3})}{\partial x^{\frac {4}{3}}_{1}\partial x^{\frac{2}{3}}_{2}\partial x^{\frac {1}{3}}_{3}}+u(x_{1},t_{1},x_{2},t_{2},x_{3},t_{3})=1 $$

together with initial and boundary conditions

$$\begin{aligned} \frac{\partial^{\frac{-3}{3}}u(x_{1},t_{1},0,t_{2},0,t_{3})}{\partial x^{\frac{-2}{3}}_{2}\, \partial x^{\frac{-1}{3}}_{3}} =& \frac{\partial ^{\frac{-5}{3}}u(0,t_{1},x_{2},t_{2},0,t_{3})}{\partial x^{\frac {-4}{3}}_{1}\, \partial x^{\frac{-1}{3}}_{3}}= \frac{\partial ^{-2}u(0,t_{1},0,t_{2},x_{3},t_{3})}{\partial x^{\frac {-4}{3}}_{1}\, \partial x^{\frac{-2}{3}}_{2}} \\ =&\frac{\partial^{\frac {-1}{3}}u(x_{1},t_{1},x_{2},t_{2},0,t_{3})}{\partial x^{\frac {-1}{3}}_{3}} =\frac{\partial^{\frac {-4}{3}}u(0,t_{1},x_{2},t_{2},x_{3},t_{3})}{\partial x^{\frac {-4}{3}}_{1}} \\ =&\frac{\partial^{\frac {-2}{3}}u(x_{1},t_{1},0,t_{2},x_{3},t_{3})}{\partial x^{\frac {-2}{3}}_{2}} = \frac{\partial^{\frac {-7}{3}}u(0,t_{1},0,t_{2},0,t_{3})}{\partial x^{\frac {4}{3}}_{1}\, \partial x^{\frac{2}{3}}_{2}\, \partial x^{\frac{1}{3}}_{3}}=0. \end{aligned}$$

Then, by taking three-dimensional Laplace transforms we obtain

$$ s^{\frac{4}{3}}_{1}s^{\frac{2}{3}}_{2}s^{\frac {1}{3}}_{3}U(s_{1},t_{1},s_{2},t_{2},s_{3},t_{3})+U(s_{1},t_{1},s_{2},t_{2},s_{3},t_{3})= \frac {1}{s_{1}s_{2}s_{3}}. $$

Then it follows that

$$ U(s_{1},t_{1},s_{2},t_{2},s_{3},t_{3})= \frac {1}{(s_{1}s_{2}s_{3})(s^{\frac{4}{3}}_{1}s^{\frac{2}{3}}_{2}s^{\frac {1}{3}}_{3}+1)}. $$


$$\begin{aligned}& L_{3} \biggl\{ \frac{\partial^{\frac {7}{3}}u(x_{1},t_{1},x_{2},t_{2},x_{3},t_{3})}{\partial x^{\frac {4}{3}}_{1}\, \partial x^{\frac{2}{3}}_{2}\, \partial x^{\frac {1}{3}}_{3}} \biggr\} \\& \quad = s^{\frac{4}{3}}_{1}s^{\frac{2}{3}}_{2}s^{\frac {1}{3}}_{3}U(s_{1},t_{1},s_{2},t_{2},s_{3},t_{3})-s^{\frac{4}{3}}_{1} L_{1} \biggl\{ \frac{\partial^{\frac {-3}{3}}u(x_{1},t_{1},0,t_{2},0,t_{3})}{\partial x^{\frac {-2}{3}}_{2}\, \partial x^{\frac{-1}{3}}_{3}} \biggr\} \\& \qquad {} - s^{\frac{2}{3}}_{2}L_{1} \biggl\{ \frac{\partial^{\frac {-5}{3}}u(0,t_{1},x_{2},t_{2},0,t_{3})}{\partial x^{\frac {-4}{3}}_{1}\, \partial x^{\frac{-1}{3}}_{3}} \biggr\} -s^{\frac {1}{3}}_{3}L_{1} \biggl\{ \frac{\partial ^{-2}u(0,t_{1},0,t_{2},x_{3},t_{3})}{\partial x^{\frac {-4}{3}}_{1}\, \partial x^{\frac{-2}{3}}_{2}} \biggr\} \\& \qquad {} -s^{\frac {4}{3}}_{1}s^{\frac{2}{3}}_{2}L_{2} \biggl\{ \frac{\partial^{\frac {-1}{3}}u(x_{1},t_{1},x_{2},t_{2},0,t_{3})}{\partial x^{\frac {-1}{3}}_{3}} \biggr\} - s^{\frac{4}{3}}_{1}s^{\frac{1}{3}}_{3}L_{2} \biggl\{ \frac{\partial^{\frac {-2}{3}}u(x_{1},t_{1},0,t_{2},x_{3},t_{3})}{\partial x^{\frac {-2}{3}}_{2}} \biggr\} \\& \qquad {} -s^{\frac{2}{3}}_{2}s^{\frac {1}{3}}_{3}L_{2} \biggl\{ \frac{\partial^{\frac {-4}{3}}u(0,t_{1},x_{2},t_{2},x_{3},t_{3})}{\partial x^{\frac {-4}{3}}_{1}} \biggr\} - \biggl\{ \frac{\partial^{\frac {-7}{3}}u(0,t_{1},0,t_{2},0,t_{3})}{\partial x^{\frac {4}{3}}_{1}\, \partial x^{\frac{2}{3}}_{2}\, \partial x^{\frac {1}{3}}_{3}} \biggr\} . \end{aligned}$$

Now, by taking the inverse three-dimensional Laplace transforms of the above equation we can get:

$$\begin{aligned} u(x_{1},t_{1},x_{2},t_{2},x_{3},t_{3})= \bigl(1-E_{\frac{4}{3}}\bigl(-x^{\frac {4}{3}}_{t_{1}}\bigr)\bigr) \bigl(1-E_{\frac{2}{3}}\bigl(-x^{\frac {2}{3}}_{t_{2}}\bigr)\bigr) \bigl(1-E_{\frac{1}{3}}\bigl(-x^{\frac{1}{3}}_{t_{3}}\bigr)\bigr). \end{aligned}$$

Example 4

As a more general example, consider

$$\begin{aligned}& \sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}}-\prod ^{n}_{i=1}\lambda _{i}R_{i}u(x_{j1},t_{j1},x_{j2},t_{j2}, \ldots,x_{jn},t_{jn}) \\& \quad =\sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}m_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial t^{m_{i}}_{ji}}. \end{aligned}$$

Then, by taking multidimensional Laplace transform to the above equation we get

$$\begin{aligned}& \sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}}-\prod ^{n}_{i=1}R_{i}(\lambda _{i}+s_{i})u(x_{j1},s_{j1},x_{j2},s_{j2}, \ldots,x_{jn},s_{jn}) \\& \quad =R_{i}u(x_{j1},0,x_{j2},0, \ldots,x_{jn},0). \end{aligned}$$

By transformation the above equation can be written as

$$ \sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}}-\prod ^{n}_{i=1}\tau _{i}u(x_{j1},s_{j1},x_{j2},s_{j2}, \ldots,x_{jn},s_{jn})=0, $$

where \(t_{ji}\) is the variable of Laplace for the time-component. Let

$$u(x_{j1},t_{j1},x_{j2},t_{j2}, \ldots,x_{jn},t_{jn})=u(x_{j1},x_{j2}, \ldots,x_{jn}). $$

Then the equation becomes

$$ \sum^{\infty}_{j=1}\frac{\partial^{\sum ^{n}_{i=1}v_{i}}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}}-\prod ^{n}_{i=1}\tau _{i}u(x_{j1},x_{j2}, \ldots,x_{jn})=0. $$

By applying the Laplace operator we can get:

$$\begin{aligned}& L_{n} \bigl\{ u(p_{j1},p_{j2}, \ldots,p_{jn}) \bigr\} \\& {\quad =\frac{\sum^{n-1}_{k=1}(1)^{n-k}\sum^{n}_{\substack{{i_{1},i_{2},\ldots,i_{k}=1}\\{i_{1},i_{2},\ldots,i_{n}}}}\prod^{k}_{j=1}s^{m_{ij}}_{ij}\prod^{n}_{\substack{i=1\\ i\neq i_{j}\\ \text{for each }j=1,2,\ldots,k}}\sum^{m_{i}}_{l_{1}=1}s^{l_{1}-1}_{i} }{\prod^{n}_{i=1}s^{m_{i}}_{i}L_{n} \{\frac{\partial^{\sum ^{n}_{i=1}-(m_{i}-v_{i})}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-v_{i})}_{i}} \}-1}} \\& {\qquad {}\times L_{k} \Biggl\{ \frac{\partial^{\sum^{n}_{\substack{{i=1}\\{i\neq i_{j}}}}(m_{i}-l_{i}+v_{i})}u(0,t_{j1},0,t_{j2}, \ldots,0,t_{jn},x_{i1},t_{ij1},0, \ldots,0,x_{i2},t_{ij2},0,\ldots,0,x_{ik},t_{ijk},0, \ldots,0 )}{\prod^{n}_{\substack{{i=1}\\{i\neq i_{j}}}}(m_{i}-l_{i}+v_{i})} \Biggr\} } \\& \qquad {}+\frac{(-1)^{n}\prod^{n}_{i=1}\sum^{m_{i}}_{l_{i}=1}s^{l_{i}-1}_{i}\frac {\partial^{-\sum ^{n}_{i=1}(m_{i}-l_{i}+v_{i})}u(0,t_{j1},0,t_{j2},\ldots,0,t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-l_{i}+v_{i})}_{i}}}{\prod^{n}_{i=1}s^{m_{i}}_{i}L_{n} \{\frac{\partial^{\sum ^{n}_{i=1}-(m_{i}-v_{i})}u(x_{j1},t_{j1},x_{j2},t_{j2},\ldots,x_{jn},t_{jn})}{\prod^{n}_{i=1}\partial x^{-(m_{i}-v_{i})}_{i}} \}-1}, \end{aligned}$$


$$ =\frac{1}{n!}L_{n} \Biggl\{ \prod^{n}_{i=1} \Biggl(x^{\alpha_{i}n+\alpha _{i}-j}_{i} \Biggl(\sum^{\infty}_{j=1} \frac{\partial^{\sum ^{n}_{i=1}v_{i}}}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}} \Biggr)E_{\alpha_{i}-\beta_{i},\alpha_{i}n\beta_{i}+1-j} \bigl(\mu_{i} x^{\alpha_{i}-\beta_{i}} \bigr) \Biggr) \Biggr\} , $$


$$ \Biggl(\sum^{\infty}_{j=1}\frac{\partial^{\sum^{n}_{i=1}v_{i}}}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}} \Biggr)E_{\alpha_{i},\beta _{i}(xji)}=\prod^{n}_{i=1} \Biggl(\sum^{\infty}_{j=0}\frac{\Gamma (n+j+1)}{\Gamma(n\alpha_{i}+\beta_{i}+\alpha ji)} \frac {x^{ji}_{i}}{j!} \Biggr). $$


$$ =\sum^{\infty}_{n=0}\frac{\prod^{n}_{i=1}\tau^{ni}_{i}}{n!} \Biggl\{ \prod^{n}_{i=1} \Biggl(x^{\alpha_{i}n+\alpha_{i}-j}_{i} \Biggl(\sum^{\infty }_{j=1}\frac{\partial^{\sum^{n}_{i=1}v_{i}}}{\prod^{n}_{i=1}\partial x^{v_{i}}_{ji}} \Biggr)E_{\alpha_{i}-\beta_{i},\alpha_{i}n\beta _{i}+1-j} \bigl(\mu_{i} x^{\alpha_{i}-\beta_{i}} \bigr) \Biggr) \Biggr\} . $$


$$ y(x)=\sum^{2}_{i=1}h_{i}y_{i}(x). $$

5 Conclusion

This work involved an implementation of the multidimensional Laplace transform to solve the multidimensional space-time fractional differential equation. Space-time derivatives of fractional order can be defined as the originators of semigroups to calculate the Fourier-Laplace signs. Hence, by the fractional calculus we established the space-time fractional diffusion equation, which is also applicable in this field. It was shown that the present method is a powerful and effective scheme for solving other fractional problems.


  1. Caputo, M: Linear models of dissipation whose Q is almost frequency independent: part II. Geophys. J. Int. 13(5), 529-539 (1967)

    Article  MathSciNet  Google Scholar 

  2. Cloot, A, Botha, J: A generalised groundwater flow equation using the concept of non-integer order derivatives. Water SA 32(1), 55-78 (2006)

    Google Scholar 

  3. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  4. Kılıçman, A, Al-Zhour, Z: Kronecker operational matrices for fractional calculus and some applications. Appl. Math. Comput. 187(1), 250-265 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Atanackovic, T, Pilipovic, S: Hamilton’s principle with variable order fractional derivatives. Fract. Calc. Appl. Anal. 14(1), 94-109 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Francesco, M, Rudolf, G: Time fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 10(3), 269-308 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Blumen, A, Zumofen, G, Klafter, J: Transport aspects in anomalous diffusion: Levy walks. Phys. Rev. A 40, 3964-3973 (1989)

    Article  Google Scholar 

  8. Chaves, A: Fractional diffusion equation to describe Levy flights. Phys. Lett. A 239, 13-16 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bhrawy, AH, Abdelkawy, MA, Alzahrani, AA, Baleanu, D, Alzahrani, EO: A Chebyshev-Laguerre-Gauss-Radau collocation scheme for solving a time fractional sub-diffusion equation on a semi-infinite domain. Proc. Rom. Acad., Ser. A 16(4), 490-498 (2015)

    MathSciNet  Google Scholar 

  10. Bhrawy, AH: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algorithms 73(1), 91-113 (2016). doi:10.1007/s11075-015-0087-2

    MathSciNet  MATH  Google Scholar 

  11. Bhrawy, AH: A new spectral algorithm for a time-space fractional partial differential equations with sub-diffusion and super-diffusion. Proc. Rom. Acad., Ser. A 17(1), 39-47 (2016)

    MathSciNet  Google Scholar 

  12. Bhrawy, AH, Baleanu, D: A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219-233 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bhrawy, AH, Zaky, MA, Baleanu, D: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2), 340-349 (2015)

    Google Scholar 

  14. Mainardi, F, Luchko, Y, Pagnini, G: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153-192 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, H, Liu, F: The fundamental solutions of the space, space-time Riesz fractional partial differential equations with periodic conditions. Numer. Math. J. Chinese Univ., Engl. Ser. 16(2), 181-192 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Ebadian, A, Rahmani, F, Khajehnasiri, A: Solution of nonlinear fractional diffusion-wave equation by triangular functions. SeMA J. 72(1), 37-46 (2015)

    Article  MATH  Google Scholar 

  17. Wijker, J: Fokker-Planck-Kolmogorov method or diffusion equation method. In: Random Vibrations in Spacecraft Structures Design. Series Solid Mechanics and Its Applications, vol. 165, pp. 323-403 (2009)

    Chapter  Google Scholar 

  18. Brooks, H, Michel, K, Amnon, A, Ora, E: Effect of inversion symmetry on the incommensurate order in multiferroic \(R\mathrm{Mn}_{2}\mathrm{O}_{5}\) (\(R=\mbox{rare earth}\)). Phys. Rev. B 78, Article ID 014407 (2008)

    Article  Google Scholar 

  19. Ahmet, B: The tanh-coth method combined with the Riccati equation for solving non-linear equation. Chaos Solitons Fractals 40, 1467-1474 (2009)

    Article  MATH  Google Scholar 

  20. Mark, M, Hans, P: Triangular array limits for continuous time random walks. Stoch. Process. Appl. 118(9), 1606-1633 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, GC, Baleanu, D, Xie, HP: Riesz Riemann-Liouville difference on discrete domains. Chaos 26, Article ID 084308 (2016). doi:10.1063/1.4958920

    Article  MathSciNet  Google Scholar 

  22. Wu, GC, Baleanu, D, Xie, HP, Chen, FL: Chaos synchronization of fractional chaotic maps based on the stability condition. Phys. A, Stat. Mech. Appl. 460, 374-383 (2016)

    Article  MathSciNet  Google Scholar 

  23. Wu, GC, Baleanu, D, Deng, ZG, Zeng, SD: Lattice fractional diffusion equation in terms of a Riesz-Caputo difference. Phys. A, Stat. Mech. Appl. 438, 335-339 (2015)

    Article  MathSciNet  Google Scholar 

  24. Wu, GC, Baleanu, D: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102, 96-99 (2014)

    Article  Google Scholar 

  25. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  26. Bochner, S: Diffusion equation and stochastic processes. Proc. Natl. Acad. Sci. USA 35(7), 368-370 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  27. Barkai, E: Fractional Fokker-Planck equation, solution and application. Phys. Rev. E 63, Article ID 046118 (2001)

    Article  Google Scholar 

  28. Murry, R: Laplace Transforms Theory and Problems, New York (1965)

Download references


The authors would like to thank the reviewers for careful reading the paper and giving constructive comments, which improved the manuscript substantially.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Wasan Ajeel Ahmood.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors jointly worked on deriving the results and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmood, W.A., Kılıçman, A. On some applications of the space-time fractional derivative. Adv Differ Equ 2016, 288 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: