 Research
 Open Access
 Published:
Analytic solution of homogeneous timeinvariant fractional IVP
Advances in Difference Equations volume 2018, Article number: 143 (2018)
Abstract
In this work, we propose a novel analytical solution approach for solving a general homogeneous timeinvariant fractional initial value problem in the normal form
where \(D_{t}^{\alpha }\) is the Caputo fractional operator with \(0<\alpha \leq 1\). The solution is given analytically in the form of a convergent multifractional power series without using any particular treatments for the nonlinear terms. The new approach is taken to search patterns for compacton solutions of several nonlinear timefractional dispersive equations, namely \(K_{\alpha }(2,2)\), \(ZK_{\alpha }(2,2)\), \(DD_{\alpha }(1,2,2)\), and \(K_{\alpha }(2,2,1)\). Remarkably, the graphical analysis showed that the nterm approximate memory solutions, labeled by the memory parameter \(0<\alpha \leq 1\), are continuously homotopic as they reflect, in some sense, some memory and heredity properties.
Introduction
The importance of fractional differential equations (FDEs) is stimulated by the appearance of many scientific models that have a nonlocal dynamical property. It has been observed that such models possess a continuum flow due to the fractional derivative effect that can be characterized by a longterm memory in time (memory index). This memory index can be interpreted as a chaotic or bifurcation behavior of a certain phenomenon for a short time subject to past circumstances. For example, it has been observed that the universal electromagnetic, acoustic, and mechanical responses are influenced by a remnant memory which can be accurately modeled by the fractional diffusion wave equations [1]. Also, it has been shown that the propagation of mechanical diffusive waves in viscoelastic media can be identified by the fractional wave equation [2]. More applications that relate to Newtonian mechanics, electromagnetism, quantum mechanics, electrochemistry, signal and image processing, and biomedical engineering can be found in [3–10].
In general, it is not an easy task to extract an analytical solution for nonlinear fractional differential equations. Almost all attempts were developed by either finding numerical solutions over a specific range or considering few terms of an iterative computational series solution as an approximate. Such available methods are (He’s) variational iteration methods [11, 12], the iterative Laplace transform method [13], Adomian’s decomposition method [14], the differential transform method [15, 16], homotopy analysis/perturbation methods [17, 18], the homotopy analysis—Laplace transform method [19], Chebyshev/Jacobi/Legendre operational matrix methods [20], the fractional Lie group method [21], the generalized Taylor power series method [22, 23], and the residual power series method [24–28].
The motivation of the current work is to explore a closedform solution of a general homogeneous timeinvariant fractional initial value problem in the normal form,
where \(D_{t}^{\alpha }\) is the Caputo fractional operator with \(\alpha \in (0,1]\), \(u(\overline{x},t)\) is an unknown function, F is an analytic differential operator in the variables \(\overline{x}= \langle x_{1},\ldots ,x_{m}\rangle \) that involves both linear and nonlinear terms, \(R\in \mathbb{R}\), and \(f(\overline{x})\in \mathcal{C}^{\infty } ( \mathbb{R}^{m} ) \). Classes of these equations include, but are not limited to, the Schrödinger equation, Korteweg–de Vries equation, Burgers–Fisher equation, Cauchy reaction–diffusion equation, Boussinesq equation, and Sharma–Tasso–Olver equation. In particular, we have successfully applied the present approach to finding closedform solutions for various nonlinear timefractional dispersive equations, namely \(K_{\alpha }(2,2)\), \(ZK_{\alpha }(2,2)\), \(DD_{\alpha }(1,2,2)\), and \(K_{\alpha }(2,2,1)\) equations. It should be pointed out here that the existence of mild solutions of the nonlocal problem of timefractional evolution equations is extensively studied in [29–38].
The organization of the current paper is as follows: In Sect. 2, we recall some necessary definitions and theorems regarding the fractional derivative and fractional power series. In Sect. 3, the solution to a general timeinvariant fractional IVP is constructed with some related convergence results and error analysis. In Sect. 4, our approach is applied to various nonlinear timefractional dispersive equations with graphical and numerical analysis to illustrate the adequacy of the proposed approach. Finally, some concluding remarks are given in Sect. 5.
Preliminaries
Many definitions and studies of fractional derivatives have been proposed in the literature. Probably this is due to the fact that no harmonious definition preserves all properties of the classical integerorder derivative. These definitions include the Grunwald–Letnikov, Riemann–Liouville, Weyl, Riesz and Caputo versions. However, in the Caputo case, the derivative of a constant function is zero and one can properly define the initial conditions for the fractional differential equations which can be handled by using an analogy with the classical integerorder case. For these reasons, we adopt the Caputo fractional derivative definition in this work.
Definition 2.1
A real function \(u(t)\), \(t>0\) is in the space \(\mathcal{C}_{\lambda \in \mathbb{R}}^{1}\) if there exists a real number \(a>\lambda \) such that \(u(t)=t^{a}v(t)\) where \(v(t)\in \mathcal{C}[0,\infty )\), and it is in the space \(\mathcal{C}_{\lambda }^{n}\) if \(u^{(n)}\in \mathcal{C}_{\lambda }^{1}\), \(n\in \mathbb{N}\).
Definition 2.2
The Riemann–Liouville fractional integral operator of order \(\alpha > 0\) associates with a real function \(u(t)\in \mathcal{C}_{ \lambda \geq 1}^{1}\) is defined as \(\mathcal{J}^{\alpha }_{t} [ u(t) ] =\frac{1}{ \Gamma (\alpha )}\int_{0}^{t} ( t\tau ) ^{\alpha 1}u( \tau )\,d\tau \), and \(\mathcal{J}^{0}_{t}\) is an identity operator.
Definition 2.3
The Caputo timefractional derivative of order \(\alpha >0\) of \(u(t)\in \mathcal{C}_{1}^{n}\), \(n\in \mathbb{N}\) is defined as \(D^{\alpha }_{t} [ u(t) ] =\mathcal{J}^{n\alpha }_{t} [ u ^{(n)}(t) ] \) if \(n1<\alpha <n\) and \(D^{\alpha }_{t} [ u(t) ] =u ^{(n)}(t)\) if \(\alpha =n\). Similarly, for n being the smallest integer that exceeds α, the Caputo timefractional derivative operator of order α is given as \(D^{\alpha }_{t} [ u(\overline{x},t) ] = \mathcal{J}^{n\alpha }_{t} [ \frac{\partial^{n}u(\overline{x},t)}{ \partial t^{n}} ] \) if \(n1<\alpha <n\) and \(D^{\alpha }_{t} [ u( \overline{x},t) ] =\frac{\partial^{n}u(\overline{x},t)}{\partial t^{n}}\) if \(\alpha =n\).
Remark 1
A direct implementation of the Caputo derivative yields \(D^{\alpha } _{t} [ t^{p} ] =\frac{\Gamma (p+1)}{\Gamma (p\alpha +1)} t ^{p\alpha }\) for \(p>0\) and \(D^{\alpha }_{t} [ c ] =0\) where c is a constant. Also, it is easy to see that the Caputo derivative is a left inverse of the Riemann–Liouville integral but not a right inverse. Specifically, for \(n1<\alpha \leq n\), \(n \in \mathbb{N}\), and \(u(t)\in \mathcal{C}_{\lambda \geq 1}^{n}\) we have \(D^{\alpha }_{t} \mathcal{J}^{\alpha }_{t} [ u(t) ] =u(t)\) and \(\mathcal{J}^{ \alpha }_{t}D^{\alpha }_{t} [ u(t) ] =u(t)\sum_{k=0}^{n1}u ^{(k)}(0^{+})\frac{t^{k}}{k!}\), where \(t>0\).
It should be noted here that it suffices to consider the Caputo fractional derivative of order \(0<\alpha \leq 1\) since \(D^{\alpha } _{t} [ u(t) ] =D^{\alpha (n1)}_{t} [ u^{(n1)}(t) ] \) for arbitrary order \(n1<\alpha \leq n\), where \(\alpha (n1)\in (0,1]\).
Definition 2.4
([25])
A fractional power series (FPS) expansion is an infinite series about \(t=t_{0}\) of the form \(\sum_{k=0}^{\infty }c_{k}(tt_{0})^{k \alpha }\) where \(0\leq n1 <\alpha \leq n\), \(t \geq t_{0}\).
Theorem 2.5
[25] Suppose that \(u(t)\) has a FPS expansion about \(t_{0}\) as above for \(t_{0}\leq t\leq t_{0}+R\). If the \(D_{t}^{k \alpha } [ u(t) ] \) are continuous on \((t_{0}, t_{0}+R)\) for \(k\in \mathbb{N}^{*}\), then \(c_{k}=\frac{D_{t}^{k\alpha } [ u(t _{0}) ] }{\Gamma (k\alpha +1)}\) where \(D_{t}^{k\alpha }\) is the kfold Caputo derivative and R is the radius of convergence.
Definition 2.6
([24])
A power series of the form
where \(\overline{x}\in I=I_{1}\times \cdots \times I_{m}\subset \mathbb{R}^{m}\), \(0<\alpha \leq 1\), and \(t\geq 0\) is called a multifractional power series about \(t=0\).
Theorem 2.7
([24])
Suppose that \(u(\overline{x},t)\) has a multifractional power series representation about \(t=0\) as above for \(\overline{x} \in I\) and \(0\leq t\leq R\). If \(D_{t}^{k\alpha } [ u(\overline{x},t) ] \) are continuous on \(I\times (0,R)\) for each \(k\in \mathbb{N}^{\ast }\), then \(f_{k}( \overline{x})=\frac{D_{t}^{k\alpha } [ u(\overline{x},0) ] }{ \Gamma (k\alpha +1)}\) where R is the radius of convergence.
Analytic solution of homogeneous timeinvariant fractional IVP
As our approach depends mainly on constructing an analytical solution of the timefractional differential equation under consideration, we first present, in a similar fashion to the classical power series, some essential convergence theorems pertaining to our proposed solution.
Theorem 3.1
Let \(\{f_{k}(\overline{x})\}_{k=0}\) be a sequence of functions \(f_{k}:I\rightarrow \mathbb{R}\). If (2.1) is convergent for some \(t=t_{0}>0\), then it is convergent for all \(t\in (0,t_{0})\).
Proof
Assume that (2.1) is convergent for \(t=t_{0}>0\). Then, for fixed \(\epsilon_{0}>0\), there exists \(N\in \mathbb{N}\) such that \(\vert f_{k}(\overline{x})t_{0}^{\alpha k} \vert <\epsilon_{0}\) for all \(k\geq N\). It follows that if \(k\geq N\), we have \(\vert f_{k}(\overline{x})t^{\alpha k} \vert <\epsilon_{0} ( \frac{t}{t_{0}} ) ^{\alpha k}\) for all \(\overline{x}\in I\) and \(t\in (0,t_{0}) \), which shows that \(\sum_{k=0}^{\infty }f_{k}(\overline{x}) t^{k\alpha }\) is absolutely convergent (and so convergent). □
We remark here that if \(f_{0}(\overline{x})\) is a bounded function on I, then the convergence of (2.1) at some \(t=t_{0}>0\) implies the convergence on \([0,t_{0})\).
Corollary 3.2
Let \(\{f_{k}(\overline{x})\}_{k=0}\) be a sequence of functions \(f_{k}:I\rightarrow \mathbb{R}\). If (2.1) is divergent for some \(t=t_{0}>0\), then it is divergent for all \(t>t_{0}\).
Proof
Suppose not. That is, (2.1) is convergent for some \(t>t_{0}\). Then from Theorem 3.1, it converges on \((0,t)\) and thus converges at \(t_{0}\), which is a contradiction. □
Corollary 3.3
Let \(\{f_{k}(\overline{x})\}_{k=0}\) be a sequence of functions \(f_{k}:I\rightarrow \mathbb{R}\). Then one of the following cases is true as regards (2.1):
 \(p_{1}\)::

The series converges only at \(t=0\);
 \(p_{2}\)::

the series converges for all \(t\geq 0\);
 \(p_{3}\)::

there exists \(R>0\) (called the radius of convergence) such that (2.1) converges for all \(t\in (0,R)\) and diverges for all \(t>R\).
Proof
Suppose it is the case that both \(p_{1}\) and \(p_{2}\) are not valid. Then there exist \(\alpha , \beta \in \mathbb{R}^{+}\) such that (2.1) converges at \(t=\alpha \) and diverges at \(t=\beta \). Therefore, the set \(T=\{t>0:\sum_{k=0}^{\infty }f_{k}(\overline{x}) t^{k\alpha } \text{ converges}\}\) is nonempty and \(T\subseteq (0,\alpha )\) by Theorem 3.1. Thus \(R:=\text{sup} T\) exists. Now, if \(t>R\), then (2.1) is divergent and if \(0< t< R\), then, by the definition of the supremum there exists \(t< t_{0}\in T\) such that (2.1) is convergent at \(t_{0}\) and so by Theorem 3.1 convergent on \((0,t_{0})\). The other cases can be handled easily. □
Now, consider the following general homogeneous timeinvariant fractional initial value problem:
where \(D_{t}^{\alpha }\) is the Caputo fractional operator with \(\alpha \in (0,1]\), \(u(\overline{x},t)\) is an unknown function, F is an analytic differential operator in the variables \(\overline{x}= \langle x_{1},\ldots ,x_{m}\rangle \) that involves both linear and nonlinear terms, \(R\in \mathbb{R}\), and \(f(\overline{x})\in \mathcal{C}^{\infty } ( \mathbb{R}^{m} ) \). In our next theorem, we exhibit a parallel scheme of the Taylor series method to solve problem (3.1). The method gives an analytical solution in the form of convergent multifractional power series without the need for linearization, perturbation, or discretization of the variables. Instead of equating terms with the same degree of homogeneity, our approach depends recursively on timefractional differentiation to obtain the unknown series coefficients.
Notation 3.4
We denote the coefficient extraction operator for a multifractional power series \(G (\overline{x},t)\), which extracts a constant multiple of the coefficient of \(t^{n\alpha }\) in G, by \([t^{n\alpha } ]_{G}\). More precisely, for \(n\geq 1\)
Note that, for a multifractional power series representation, \(G (\overline{x},t)=\sum_{k=0}^{\infty }g_{k}(\overline{x}) t^{k \alpha }\), we have
where \(D_{t}^{n\alpha }=D_{t}^{\alpha }\cdots D_{t}^{\alpha }\) (n times).
Theorem 3.5
Suppose that the solution \(u(\overline{x},t)\) of (3.1) has a convergent multifractional power series representation of the form (2.1) for \(\overline{x}\in I\subset \mathbb{R}^{m}\), and \(0\leq t\leq R\). If \(D_{t}^{n\alpha } [ u(\overline{x},t) ] \) are continuous on \(I\times (0,R)\) for \(n\in \mathbb{N}\), then the solution of (3.1) is given analytically by
Proof
As \(u(\overline{x},t)\) satisfies the initial condition, we should have \(f_{0}(\overline{x})=f(\overline{x})\). Applying the operator \(D_{t}^{\alpha }\) to Eq. (3.1) \((n1)\) times, using the linearity property of the Caputo operator, and Remark 1 we have for \(n\geq 1\)
for all \(\overline{x}\in I\) and \(0\leq t\leq R\). In particular at \(t=0\), we obtain
for \(\overline{x}\in I\), and hence
as required. □
As an immediate consequence of Theorems 2.7 and 3.5, we obtain the following generalized Taylor formula.
Corollary 3.6
Let \(u(\overline{x},t)\) be the solution of (3.1) under the same hypotheses of Theorem 3.5. Then \(u(\overline{x},t)\) is analytic at \(t=0\) in the sense of fractional power series. I.e.
As an immediate special case of Theorem 3.5, we have the following explicit description of the solution coefficients in terms of the previous coefficient.
Corollary 3.7
If F is a linear differential operator with constant coefficients, i.e. \(F(u)=\sum_{j=1}^{m}\sum_{i=0}^{k}a_{ij}\frac{\partial^{i}u}{ \partial x_{j}^{i}}\) where \(a_{ij}\)’s are constants, then for \(n\geq 1\)
Remark 2
In practical terms to find \(f_{n}(\overline{x})\) for \(n\geq 1\), it is sufficient to substitute the \((n1)\)truncated series \(\sum_{k=0}^{n1}f _{k}(\overline{x})t^{k\alpha }\) of \(u(\overline{x},t)\) in Eq. (3.7) since the remaining terms contain higher powers \(\mathcal{O}(t^{n\alpha })\) and \(D_{t}^{(n1)\alpha } [\mathcal{O}(t ^{n\alpha }) ]= \mathcal{O}(t)=0\) when \(t=0\).
Due to the complexity related to this kind of equations, it is not always possible to find a general expression for the series coefficients. In such a case, the solution can be found approximately as a partial sum of the series, \(u_{n}(\overline{x},t)=\sum_{k=0}^{n}f _{k}(\overline{x}) t^{\alpha k}\) in some reasonable interval of t, and thus the overall errors can be made smaller by adding more new terms as shown in the following case.
Corollary 3.8
Let \(\{f_{k}(\overline{x})\}_{k=0}\) be a uniformly bounded sequence of functions \(f_{k}:I\rightarrow \mathbb{R}\). Then (2.1) is uniformly convergent for \(0\leq t \leq R<1\). Moreover, \(\Vert u(\overline{x},t)u_{n}(\overline{x},t) \Vert \rightarrow 0\) as \(n\rightarrow \infty \).
Proof
By the uniform boundedness of \(\{f_{k}(\overline{x})\}_{k=0}\), there exists \(M>0\) such that \(\vert f_{k}(\overline{x}) \vert \leq M\) for all \(k\in \mathbb{N}\) and \(\overline{x}\in I\). Thus \(\vert f_{k}(\overline{x})t^{\alpha k} \vert \leq MR^{\alpha k}\). Since \(\sum_{k=0} ^{\infty }MR^{\alpha k}\) is convergent geometric series (with ratio \(R^{\alpha }<1\)), by the comparison test, we see that \(\sum_{k=0}^{ \infty }f_{k}(\overline{x}) t^{\alpha k}\) is uniformly convergent as desired. Moreover,
□
Applications and discussions
The goal of this section is to verify the applicability of our proposed approach, derived from Theorem 3.5, in studying the memory effects due to the timefractional derivative on various nonlinear dispersive equations. It should be noted here that all necessary calculations and graphical analysis have been done by using Mathematica 10 software.
Example 1
In [39], Rosenau and Hyman introduced a class of solitary waves with compact support (called compactons) to understand the role of nonlinear dispersion on pattern formation in liquid drops. These compactons are solitary waves free of exponential tails and they reemerge with exactly the same coherent shape after a collision. Motivated by the importance of taking into account the memory effects due to timefractional derivative, we first consider the timefractional version \(K_{\alpha }(2,2)\) of the thirdorder Rosenau–Hyman equation, which reads
subject to the initial condition with compact support
where \(t\geq 0\), \(a\in \mathbb{R}^{+}\), and \(0<\alpha \leq 1\). In this case we have \(F(u)= (a(u^{2})_{x}+(u^{2})_{xxx} )\). In accordance with Theorem 3.5, the proposed fractional power series solution of Eqs. (4.1) and (4.2) has the form
Following Eq. (3.7) and taking into account Remark 2, we successively obtain the coefficients \(f_{n}(x)\) as follows:
and so on. In general, for \(n\geq 1\) we have
Therefore, in view of (3.4) we have the exact memory solution of \(K_{\alpha }(2,2)\) in closed form,
where \(E_{2\alpha ,1} ( \cdot ) \) and \(E_{2\alpha ,\alpha +1} ( \cdot ) \) are the twoparameter Mittag–Leffler functions. Similar versions of (4.1) were handled using homotopy analysis/perturbation methods [40, 41] and the reduced differential transform approach [40], where the obtained solutions were consistent with (4.5). Particularly when \(\alpha =1\), we have the exact solution for the classical \(K(2,2)\) Rosenau–Hyman equation [39]
with compact support \(\vert xct\vert \leq \frac{2\pi }{\sqrt{a}}\).
Figure 1 shows the effect of different values of the fractional derivative order \(0<\alpha \leq 1\) on the cross section approximate solutions \(u_{4}(x,t)=\sum_{n=0}^{4}f_{n}(x)t^{n\alpha }\) of the equation for \(K_{\alpha }(2,2)\) with fixed \(t=0.5\) and \(c=a=1\) in the compact support \(\vert x0.5\vert \leq 2\pi \). Clearly, \(u_{4}(x,0.5)\) for \(\alpha =1\) is in high agreement with the cross section of the exact solution for \(K(2,2)\) on its compact support. Furthermore, \(u_{4}(x,0.5)\) for different values of \(0<\alpha \leq 1\) are continuously homotopic, as α increases, to the corresponding one of the exact solution of \(K(2,2)\).
Table 1 shows the consecutive errors \(con_{n}(x,t)=\vert u_{n+1}(x,t)u_{n}(x,t) \vert \) for \(n=4\) and \(n=10\). It is clear that we have a remarkable accuracy even with few terms and we can have more accuracy when n gets larger and/or t gets closer to zero. Also, we can see that the error approaches zero as the memory index α approaches one.
Example 2
In [42, 43], Zakharov and Kuznetsov introduced an isotropic twodimensional equation \(ZK(2,2)\) to describe the weakly nonlinear ion acoustic waves in a strongly magnetized lossless plasma. As our next example, we consider the timefractional nonlinear dispersive equation \(ZK_{\alpha }(2,2)\),
subject to the initial condition with compact support
where \(t\geq 0\) and \(0<\alpha \leq 1\). From Theorem 3.5, the fractional power series solution for Eqs. (4.7) and (4.8) has the form
Proceeding as before with straightforward calculations, the coefficients are obtained recursively as
and so on. For \(n\geq 1\), we have the following general form for the unknown coefficients:
Therefore, the memory solution in closed form is given by
In particular when \(\alpha =1\), we have the exact solution to the classical \(ZK(2,2)\) equation [44]
Figure 2 shows the effect of the memory index \(0<\alpha \leq 1\) on the surface approximate solutions \(u_{4}(x,y,t)\) of \(ZK_{\alpha }(2,2)\) with fixed \(t=0.5\) and \(c=1\). In some sense, \(u_{4}(x,y,0.5)\) for different values of α are continuously communicated to reach the corresponding one when \(\alpha =1\).
Table 2 shows the consecutive errors for \(n=4\) and \(n=10\). Apparently, we have a smaller error when t is relatively small, n is large, or α approaches one.
Example 3
In [45], Rosenau proposed a dispersive dissipative entity \(DD(k,m,n)\) which combines the interaction between convection, dispersion and dissipation. In this example, we consider the timefractional version \(DD_{\alpha }(1,2,2)\) (the classical one is known as the \(K(2,2)\)Burger equation), which reads
subject to the initial condition \(u(x,0)=2 ( 1+\tanh (  \frac{x}{4} ) ) ^{1}= ( 1+e^{\frac{x}{2}} ) \), where \(t\geq 0\) and \(0<\alpha \leq 1\). Following Eq. (3.7), we recursively obtain the coefficients \(f_{n}(x)\) for \(n\geq 1\) as follows:
Therefore, in view of (3.4) we have the memory solution of \(DD_{\alpha }(1,2,2)\) in closed form,
Particularly with \(\alpha =1\), we have the exact solution \(u(x,t)= ( 1+e^{\frac{xt}{2}} ) =2 ( 1+\tanh ( \frac{tx}{4} ) ) ^{1}\) for the classical \(DD(1,2,2)\) equation [46].
Figure 3 shows the behavior of the approximate solutions \(u_{4}(x,0.5)\) of \(DD_{\alpha }(1,2,2)\) for various values of \(0<\alpha \leq 1\). Evidently, \(u_{4}(x,0.5)\) when \(\alpha =1\) is in harmony with the exact solution in \(x\in [ \frac{4\pi +1}{2},\frac{4\pi +1}{2} ] \). Moreover, \(u_{4}(x,0.5)\) for different values of α is continuously communicated until \(\alpha =1\) is reached. Thus a convenient solution is expected for various values of α.
Example 4
In [47], Dey studied the role of the fifthorder dispersion term in the existence of the compacton solutions and as regards the soliton stability for the usual Korteweg–de Vries (KdV) equation. We next consider the timefractional version \(K_{\alpha }(2,2,1)\) of the form
subject to the initial condition \(u(x,0)=\frac{16c1}{12}\cosh^{2} ( \frac{x}{4} ) \), where \(t\geq 0\) and \(0<\alpha \leq 1\). Following the formula (3.7), we recursively obtain the coefficients \(f_{n}(x)\), \(n\geq 1\) as follows:
Therefore in view of (3.4), the solution of \(K_{\alpha }(2,2,1)\) is given in closed form by
which is identical to the solution obtained by using the homotopy perturbation method [48]. Particularly with \(\alpha =1\), we have the exact solution for the classical \(K(2,2,1)\) equation [49],
with compact support \(\vert xct\vert \leq 2\pi \).
Figure 4 exhibits the approximate solutions \(u_{4}(x,0.5)\) of \(K_{\alpha }(2,2,1)\) for various values of \(0<\alpha \leq 1\). Apparently, \(u_{4}(x,0.5)\) when \(\alpha =1\) is in harmony with the exact solution on its compact support. Further, \(u_{4}(x,0.5)\) for different values of α is continuously communicated until \(\alpha =1\) is reached.
Conclusion
In this paper, a hybrid version of the power series method is presented to handle a general timeinvariant fractional initial value problem of the form (3.1). The solution of (3.1) is given analytically in the form of a convergent multifractional power series (2.1) without the need of any linearization, perturbation, or discretization methods. Several nonlinear dispersive examples were tested and the exact memory solutions were successfully obtained in closed form. In fact, the physical interpretation of these solutions is beyond the scope of this work. However, in some sense, the graphs of the nterm approximate memory solutions, labeled by the memory index \(0<\alpha \leq 1\), were continuously deformed as they represent the memory and hereditary effects.
References
 1.
Nigmatullin, R.R.: To the theoretical explanation of the “Universal response”. Phys. Status Solidi B 123, 739–745 (1984)
 2.
Mainardi, F., Paradisi, P.: Fractional diffusive waves. J. Comput. Acoust. 9, 1417–1436 (2001)
 3.
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
 4.
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
 5.
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)
 6.
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)
 7.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
 8.
Diethelm, K.: The Analysis of Fractional Differential Equations: An ApplicationOriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)
 9.
Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54, 263–282 (2008)
 10.
Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion to discretize fractional order derivatives—an expository review. Nonlinear Dyn. 38, 155–170 (2004)
 11.
Shakeri, F., Dehghan, M.: Numerical solution of the KleinGordon equation via He’s variational iteration method. Nonlinear Dyn. 51, 89–97 (2007)
 12.
Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7, 27–34 (2006)
 13.
Jafari, H., Nazari, M., Baleanu, D., Khalique, C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66, 838–843 (2013)
 14.
Ray, S.S., Bera, R.K.: Analytical solution of a fractional diffusion equation by Adomian decomposition method. Appl. Math. Comput. 174, 329–336 (2006)
 15.
Liu, J., Hou, G.: Numerical solutions of the space and timefractional coupled Burgers equations by generalized differential transform method. Appl. Math. Comput. 217, 7001–7008 (2011)
 16.
Odibat, Z., Kumar, S., Shawagfeh, N., Alsaedi, A., Hayat, T.: A study on the convergence conditions of generalized differential transform method. Math. Methods Appl. Sci. 40, 40–48 (2017)
 17.
Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2010)
 18.
He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)
 19.
Kumar, A., Kumar, S.: A modified analytical approach for fractional discrete KdV equations arising in particle vibrations. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 88, 95–106 (2018)
 20.
Bhrawy, A.H., Taha, T.M., Machado, J.A.T.: A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn. 81, 1023–1052 (2015)
 21.
Jafari, H., Kadkhoda, N., Baleanu, D.: Fractional lie group method of the timefractional Boussinesq equation. Nonlinear Dyn. 81, 1569–1574 (2015)
 22.
Alquran, M., Jaradat, I.: A novel scheme for solving Caputo timefractional nonlinear equations: theory and application. Nonlinear Dyn. 91, 2389–2395 (2018)
 23.
Jaradat, I., AlDolat, M., AlZoubi, K., Alquran, M.: Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fractals 108, 107–110 (2018)
 24.
ElAjou, A., AbuArqub, O., Momani, S.: Approximate analytical solution of the nonlinear fractional KdVBurgers equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)
 25.
ElAjou, A., AbuArqub, O., AlZhour, Z., Momani, S.: New results on fractional power series: theories and applications. Entropy 15, 5305–5323 (2013)
 26.
Kumar, A., Kumar, S., Yan, S.P.: Residual power series method for fractional diffusion equations. Fundam. Inform. 151, 213–230 (2017)
 27.
Zhang, Y., Kumar, A., Kumar, S., Baleanu, D., Yang, X.J.: Residual power series method for timefractional Schrödinger equations. J. Nonlinear Sci. Appl. 9, 5821–5829 (2016)
 28.
Alquran, M., Jaradat, H.M., Syam, M.I.: Analytical solution of the timefractional Phi4 equation by using modified residual power series method. Nonlinear Dyn. 90, 2525–2529 (2017)
 29.
Chen, P., Li, Y.: Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions. Z. Angew. Math. Phys. 65, 711–728 (2014)
 30.
Chen, P., Zhang, X., Li, Y.: Nonlocal problem for fractional stochastic evolution equations with solution operators. Fract. Calc. Appl. Anal. 19, 1507–1526 (2016)
 31.
Chen, P., Zhang, X., Li, Y.: Approximation technique for fractional evolution equations with nonlocal integral conditions. Mediterr. J. Math. 14(14), 226 (2017)
 32.
Chen, P., Zhang, X., Li, Y.: Study on fractional nonautonomous evolution equations with delay. Comput. Math. Appl. 73, 794–803 (2017)
 33.
Chen, P., Li, Y., Chen, Q., Feng, B.: On the initial value problem of fractional evolution equations with noncompact semigroup. Comput. Math. Appl. 67, 1108–1115 (2014)
 34.
Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011)
 35.
Ouyang, Z.: Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. Comput. Math. Appl. 61, 860–870 (2011)
 36.
Tatar, N.: Existence results for an evolution problem with fractional nonlocal conditions. Comput. Math. Appl. 60, 2971–2982 (2010)
 37.
Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. 71, 4471–4475 (2009)
 38.
Li, F., Liang, J., Xu, H.K.: Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, 510–525 (2012)
 39.
Rosenau, P., Hyman, J.M.: Compactons: solitons with finite wavelengths. Phys. Rev. Lett. 70, 564–567 (1993)
 40.
Singh, J., Kumar, D., Swroop, R., Kumar, S.: An efficient computational approach for timefractional Rosenau–Hyman equation. Neural Comput. Appl. 2017, 1–8 (2017)
 41.
Odibat, Z.: Compact and noncompact structures for nonlinear fractional evolution equations. Phys. Lett. A 372, 1219–1227 (2008)
 42.
Zakharov, V.E., Kuznetsov, E.A.: On threedimensional solitons. Sov. Phys. JETP 39, 285–288 (1974)
 43.
Kivshar, Y.S., Pelinovsky, D.E.: Selffocusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117–195 (2000)
 44.
Wazwaz, A.M.: Nonlinear dispersive special type of the Zakharov–Kuznetsov equation \(ZK(n,n)\) with compact and noncompact structures. Appl. Math. Comput. 161, 577–590 (2005)
 45.
Rosenau, P.: On a class of nonlinear dispersivedissipative interactions. Physica D 123, 525–546 (1998)
 46.
Wazwaz, A.M.: The tanh method for compact and noncompact solutions for variants of the KdVBurger and the \(K(n,n)\)Burger equations. Physica D 213, 147–151 (2006)
 47.
Dey, B.: Compacton solutions for a class of two parameter generalized oddorder Korteweg–de Vries equations. Phys. Rev. E 57, 4733–4738 (1998)
 48.
Koçak, H., Özis, T., Yildirim, A.: Homotopy perturbation method for the nonlinear dispersive \(K(m,n,1)\) equations with fractional time derivatives. Int. J. Numer. Methods Heat Fluid Flow 20, 174–185 (2010)
 49.
Zhu, Y., Tong, K., Chaolu, T.: New exact solitarywave solutions for the \(K(2,2,1)\) and \(K(3,3,1)\) equations. Chaos Solitons Fractals 33, 1411–1416 (2007)
Acknowledgements
The authors would like to thank the editor and the referees for their valuable time and consideration.
Author information
Affiliations
Contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Corresponding author
Correspondence to Imad Jaradat.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Received
Accepted
Published
DOI
MSC
 26A33
 35R11
 35F25
 35C10
 40C15
Keywords
 Caputo derivative
 Fractional power series
 Fractional PDE
 Power series method