Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function

Abstract

We discuss and derive the analytical solution for the fractional partial differential equation with generalized Riemann-Liouville fractional operator 𝐷^{𝛼,𝛽0,𝑡} of order 𝛼 and 𝛽. Here, we derive the solution of the given differential equation with the help of Laplace and Hankel transform in terms of Fox's 𝐻-function as well as in terms of Fox-Wright function _𝑝𝜓_𝑞.

1. Introduction, Definition, and Preliminaries

Applications of fractional calculus require fractional derivatives of different kinds [1–9]. Differentiation and integration of fractional order are traditionally defined by the right-sided Riemann-Liouville fractional integral operator 𝐼^𝑃𝑎+ and the left-sided Riemann-Liouville fractional integral operator 𝐼^𝑃𝑎−, and the corresponding Riemann-Liouville fractional derivative operators 𝐷^𝑃𝑎+ and 𝐷^𝑃𝑎−, as follows [10, 11]:𝐼^𝜇𝑎+𝑓(𝑥)=Γ(1𝜇)^𝑥𝑎𝑓(𝑡)(𝑥−𝑡)^1−𝜇𝐼𝑑𝑡(𝑥>𝑎;𝑅(𝜇)>0),(1.1)_𝜇𝑎−𝑓(𝑥)=1Γ(𝜇)^𝑎𝑥𝑓(𝑡)(𝑡−𝑥)^1−𝜇𝐷𝑑𝑡(𝑥<𝑎;𝑅(𝜇)>0),(1.2)_𝜇𝑎±𝑓(𝑥)=±𝑑𝑑𝑥^𝑛𝐼^{𝑛−𝜇𝑎±}𝑓(𝑥)(𝑅(𝜇)≥0;𝑛=[𝑅(𝜇)]+1),(1.3)

where the function 𝑓 is locally integrable, 𝑅(𝜇) denotes the real part of the complex number 𝜇∈𝐶 and [𝑅(𝜇)] means the greatest integer in 𝑅(𝜇).

Recently, a remarkable large family of generalized Riemann-Liouville fractional derivatives of order 𝛼(0<𝛼<1) and type 𝛽(0≤𝛽≤1) was introduced as follows [1–3, 5, 6, 8].

Definition 1.1. The right-sided fractional derivative 𝐷^{𝛼,𝛽𝑎+} and the left-sided fractional derivative 𝐷^{𝛼,𝛽𝑎−} of order 𝛼(0<𝛼<1) and type 𝛽(0≤𝛽≤1) with respect to 𝑥 are defined by 𝐷^{𝛼,𝛽𝑎±}𝑓(𝑥)=±𝐼^{𝛽(1−𝛼)𝑎±}𝑑𝑑𝑥𝐼^{(1−𝛽)(1−𝛼)𝑎±}𝑓(𝑥),(1.4) whenever the second number of (1.4) exists. This generalization (1.4) yields the classical Riemann-Liouville fractional derivative operator when 𝛽=0. Moreover, for 𝛽=1, it gives the fractional derivative operator introduced by Liouville [12] which is often attributed to Caputo now-a-days and which should more appropriately be referred to as the Liouville-Caputo fractional derivative. Several authors [7, 9] called the general operators in (1.4) the Hilfer fractional derivative operators. Applications of 𝐷^{𝛼,𝛽𝑎±} are given [3].
Using the formulas (1.1) and (1.2) in conjunction with (1.3) when 𝑛=1, the fractional derivative operator 𝐷^{𝛼,𝛽𝑎±} can be written in the following form: 𝐷^{𝛼,𝛽𝑎±}𝑓(𝑥)=±𝐼^{𝛽(1−𝛼)𝑎±}𝐷^{𝛼+𝛽−𝛼𝛽𝑎±}𝑓(𝑥).(1.5) The difference between fractional derivatives of different types becomes apparent from their Laplace transformations. For example, it is found for 0<𝛼<1 that [1, 2, 9] 𝐿𝐷^𝛼,𝛽0+𝑓(𝑥)(𝑠)=𝑠^𝛼𝐿[𝑓(𝑥)](𝑠)−𝑠^{𝛽(𝛼−1)}𝐼^{(1−𝛽)(1−𝛼)0+}𝑓(0+)(0<𝛼<1),(1.6) where (𝐼^{(1−𝛽)(1−𝛼)0+}𝑓)(0+) is the Riemann-Liouville fractional integral of order (1−𝛽)(1−𝛼) evaluated in the limit as 𝑡→0+, it being understood (as usual) that [13], 𝐿[𝑓(𝑥)](𝑠)∶=^∞0𝑒^−𝑠𝑥𝑓(𝑥)𝑑𝑥∶=𝐹(𝑠),(1.7) provided that the defining integral in (1.7) exists.
The familiar Mittag-Leffler functions 𝐸_𝜇(𝑧) and 𝐸_𝜇,𝜈(𝑧) are defined by the following series: 𝐸_𝜇(𝑧)∶=^∞_𝑛=0𝑧^𝑛Γ(𝜇𝑛+1)∶=𝐸^𝜇,1𝐸(𝑧)(𝑧∈𝐶;𝑅(𝜇)>0),(1.8)_𝜇,𝜈(𝑧)∶=^∞_𝑛=0𝑧^𝑛Γ(𝜇𝑛+𝜈)(𝑧,𝜈∈𝐶;𝑅(𝜇)>0),(1.9) respectively. These functions are natural extensions of the exponential, hyperbolic, and trigonometric functions, since 𝐸₁(𝑧)𝐸=𝑒^𝑧,𝐸₂𝑧²=cosh𝑧,𝐸₂−𝑧²=cos𝑧,_1,2(𝑧)=𝑒^𝑧−1𝑧,𝐸^2,2𝑧²=sin𝑧h𝑧.(1.10)
For a detailed account of the various properties, generalizations, and applications of the Mittag-Leffler functions, the reader may refer to the recent works by, for example, Gorenflo et al. [14] and Kilbas et al. [15–17]. The Mittag-Leffler function (1.1) and some of its various generalizations have only recently been calculated numerically in the whole complex plane [18, 19]. By means of the series representation, a generalization of the Mittag-Leffler function 𝐸_𝜇,𝜈(𝑧) of (1.2) was introduced by Prabhakar [20] as follows: 𝐸^{𝜆𝜇,𝜈}(𝑧)=^∞_𝑛=0(𝜆)_𝑛Γ(𝜇𝑛+𝜈)𝑧𝑛^𝑛!(𝑧,𝜈,𝜆∈𝐶;𝑅(𝜇)>0),(1.11) where (𝜆)_𝜈 denotes the familiar Pochhammer symbol, defined (for 𝜆,𝜈∈𝐶 and in terms of the familiar Gamma function) by (𝜆)_𝜈∶=Γ(𝜆+𝜈)Γ(𝜆)=1𝜆(𝜆+1)⋯(𝜆+𝑛−1)((𝜈𝜈==0𝑛;∈𝜆𝑁∈;𝐶𝜆⧵∈{𝐶0})).(1.12)
Clearly, we have the following special cases: 𝐸^1𝜇,𝜈(𝑧)=𝐸_𝜇,𝜈(𝑧),𝐸^1𝜇,1(𝑧)=𝐸_𝜇(𝑧).(1.13) Indeed, as already observed earlier by Srivastava and Saxena [21], the generalized Mittag-Leffler function 𝐸^{𝜆𝜇,𝜈}(𝑧) itself is actually a very specialized case of a rather extensively investigated function _𝑝Ψ_𝑞 as indicated below [17]: 𝐸^{𝜆𝑢,𝜈}(𝑧)=1Γ(𝜆)¹Ψ₁((𝜆𝜈,,u1));;𝑧.(1.14) Here and in what follows, _𝑝Ψ_𝑞 denotes the Wright (or more appropriately, the Fox-Wright) generalized of the hypergeometric _𝑝F_𝑞 function, which is defined as follows [12]: _𝑝Ψ_𝑞=𝑎¹𝑏₁,𝐴₁,…,𝑎_𝑝,𝐴_𝑝;,𝐵₁,…,𝑏_𝑞,𝐵_𝑞;𝑧=^∞_𝑥=0Γ𝑎₁+𝐴₁𝑘⋯Γ𝑎_𝑝+𝐴_𝑝𝑘𝑧^𝑘𝑅𝐴Γ𝑏₁+𝐵₁𝑘⋯Γ𝑏_𝑞+𝐵_𝑞𝑘𝑘!,(1.15)_𝑗>0(𝑗=1,…,𝑝);𝑅𝐵_𝑗>0(𝑗=1,…,𝑞);1+𝑅∑^𝑞_𝑗=1𝐵_𝑗−∑^𝑝_𝑗=1𝐴_𝑗≥0,(1.16) in which we assumed in general that 𝑎_𝑗,𝐴_𝑗∈𝐶(𝑗=1,…,𝑝),𝑏_𝑗,𝐵_𝑗∈𝐶(𝑗=1,…,𝑞).(1.17)
In application of Mittag-Leffler function, it is useful to have the following Laplace inverse transform formula: 𝐿⁻¹(𝑆^𝛾𝑆+𝐴^𝛾−𝛽)^𝑘+1=𝑘1!𝑡^{𝛾𝑘+𝛽−1}𝐸^{𝑘𝛾,𝛽}(−𝐴𝑡^𝛾),(1.18) where 𝐸^{𝑗𝛾,𝛽}(𝑧)=(𝑑^𝑗/𝑑𝑧^𝑗)𝐸_𝛾,𝛽(𝑧).

2. Fox’s 𝐻-function

The Fox function, also referred as the Fox’s 𝐻-function, generalizes the Mellin-Barnes function. The importance of the Fox function lies in the fact that it includes nearly all special functions occurring in applied mathematics and statistics as special cases. Fox 𝐻-function is defined as [22]𝐻^{1,𝑝𝑝,𝑞+1}⎡⎢⎢⎣−𝑥|||||(0,1)1,−1𝑎−₁,𝐴𝑏₁₁,…,1−𝑎_𝑝,𝐴_𝑝,𝐵₁,…,1−𝑏_𝑞,𝐵_𝑞⎤⎥⎥⎦=^∞_𝑘=0Γ𝑎₁+𝐴₁𝑘⋯Γ𝑎_𝑝+𝐴_𝑝𝑘𝑘!Γ𝑏₁+𝐵₁𝑘⋯Γ𝑏_𝑝+𝐵_𝑞𝑘𝑥^𝑘.(2.1)

We need this relation𝐸^{𝑘𝛼,𝛽}(𝑥)=^∞_𝑛=𝑘𝑛!𝑥^𝑛−𝑘(𝑛−𝑘)!Γ(𝛼𝑛+𝛽)=^∞_𝑗=0Γ(𝑗+𝑘+1)𝑥^𝑗=𝐻_1,11,2𝑗!Γ(𝛼𝑗+𝛼𝑘+𝛽)−𝑥||||(0,1),((1−−𝑘,𝛼1𝑘)−𝛽,𝛼).(2.2)

3. Finite Hankel Transform

If 𝑓(𝑟) satisfies Dirichlet conditions in closed interval (0,𝑎) and if its finite Hankel transform is defined to be [23]𝐻[𝑓(𝑟)]=𝑓𝜆_𝑛=^𝑎0𝑟𝑓(𝑟)𝐽₀𝑟𝜆_𝑛𝑑𝑟,(3.1)

where 𝜆_𝑛 are the roots of the equation 𝐽₀(𝑟)=0. Then at each point of the interval at which 𝑓(𝑟) is continuous:𝑓(𝑟)=2𝑎^2∞_𝑛=1𝑓𝜆_𝑛𝐽⁰𝜆_𝑛𝑟𝐽²¹𝜆_𝑛𝑎,(3.2)

where the sum is taken over all positive roots of 𝐽₀(𝑟)=0, 𝐽₀ and 𝐽₁ are Bessel functions of first kind.

In application of the finite Hankel transform to physical problems, it is useful to have the following formula [23]𝐻𝑑²𝑓𝑑𝑟²+1𝑟𝑑𝑑𝑓𝑟=−𝜆^2𝑛𝑓(𝑟)+𝑎𝜆_𝑛𝑓(𝑎)𝐽₁𝜆_𝑛𝑎.(3.3)

Example 3.1. Solve the differential equation 𝐷^{2𝛼,𝛽0,𝑡}𝑢(𝑟,𝑡)+𝑎𝐷^{𝛼,𝛽0,𝑡}𝑢(𝑟,𝑡)=𝑑𝜕²𝑢(𝑟,𝑡)𝜕𝑟²+1𝑟𝑢(𝑟,𝑡)+𝑓(𝑡),(3.4) where 0<𝛼≤1/2 and 0≤𝛽≤1
with initial condition 𝐼𝐼^{𝑡(1−𝛽)(1−2𝛼)}𝑢(𝑟,0)=𝜙₁(𝑟),_{𝑡(1−𝛽)(1−𝛼)}𝑢𝑢(𝑟𝑢(,𝑟(𝑡,𝑟)𝑡,)=𝑡)=0=fin0eivt𝑢eef(or𝑟yr,aw0t𝑟)h𝑟=e==r1e𝜙,0f₂,o𝑡(r𝑡>𝑟)𝑡>,0<,00.,(3.5)

Solution 1. Taking Laplace transform of (3.4), we get 𝑠^2𝛼𝑢̃(𝑟,𝑠)−𝑠^{𝛽(2𝛼−1)}𝜙₁(𝑟)+𝑎𝑠^𝛼𝑢̃(𝑟,𝑠)−𝑎𝑠^{𝛽(𝛼−1)}𝜙₂(𝑟)=𝑑𝜕²𝑢̃(𝑟,𝑠)𝜕𝑟²+1𝑟𝑢̃(𝑟,𝑠)+𝑓(𝑠).(3.6) Taking Hankel transform on both side of the above equation, we get 𝑠^2𝛼𝑢̃̃(𝑟,𝑠=)−𝑑𝑠−𝜆^{𝛽(2𝛼−1)}_2𝑛𝜙₁(𝑟)+𝑎𝑠^𝛼𝑢̃̃(𝑟,𝑠)−𝑎𝑠^{𝛽(𝛼−1)}𝜙₂(𝑟)𝑢̃̃(𝑟,𝑠)+𝑓(𝑠)𝐽¹𝜆_𝑛𝜆_𝑛,(3.7) then we get 𝑢̃̃(𝑟,𝑠)=𝑠^{𝛽(2𝛼−1)}𝜙₁(𝑟)𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛+𝑎𝑠^{𝛽(𝛼−1)}𝜙₂(𝑟)𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛+𝑠^2𝛼+𝑓𝑎𝑠(^𝛼𝑠)+𝑑𝜆^2𝑛𝐽¹𝜆_𝑛𝜆_𝑛𝑢̃̃(𝑟,𝑠)=𝐺₁,(3.8)𝜙₁(𝑟)+𝑎𝐺₂𝜙₂(𝑟)+𝐺₃𝑓(𝑠)𝐽¹𝜆_𝑛𝜆_𝑛,(3.9) where 𝐺₁=𝑠^{𝛽(2𝛼−1)}𝐺𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛,(3.10)₂=𝑠^{𝛽(𝛼−1)}𝐺𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛,(3.11)₃=1𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛.(3.12)
On taking Laplace inverse of (3.10), (3.11), and (3.12), respectively, 𝐿⁻¹𝑠^2𝛼+𝑠𝑎𝑠^𝛼+^{𝛽(2𝛼−1)}𝑑𝜆^2𝑛=^∞_𝑚=0(−1)^𝑚𝑎^𝑚+1𝑡^{𝛼+𝛽−2𝛼𝛽−𝑚𝛼−1}𝐿𝑚!𝐸^{𝑚𝛼,𝛼+𝛽−2𝛼𝛽−2𝑚𝛼}−𝑑𝑎𝜆^2𝑛𝑡^𝛼,(3.13)₋₁𝑠^2𝛼+𝑠𝑎𝑠^𝛼+^{𝛽(𝛼−1)}𝑑𝜆^2𝑛=^∞_𝑚=0(−1)^𝑚𝑎^𝑚+1𝑡^{𝛼+𝛽−𝛼𝛽−𝑚𝛼−1}𝐿𝑚!𝐸^{𝑚𝛼,𝛼+𝛽−𝛼𝛽−2𝑚𝛼}−𝑑𝑎𝜆^2𝑛𝑡^𝛼,(3.14)₋₁𝑠^2𝛼+𝑎𝑠1^𝛼+𝑑𝜆^2𝑛=^∞_𝑚=0(−1)^𝑚𝑎^𝑚+1𝑡^{𝛼−𝑚𝛼−1}𝑚!𝐸^{𝑚𝛼,𝛼−2𝑚𝛼}−𝑑𝑎𝜆^2𝑛𝑡^𝛼.(3.15) After taking Inverse Laplace and Hankel transform of (3.9) put the value (3.13) through (3.15) in (3.9), we get 𝑢(𝑟,𝑡)=2^∞_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝑚!𝐽²¹𝜆_𝑛𝜙¹(𝑟)𝑡^{∞−2𝛼𝛽−𝑚𝛼+𝛼+𝛽−1}_𝑗=0(𝑗+𝑚+1)!−𝑑𝜆^2𝑛𝑡^𝛼/𝑎^𝑗+2𝑎_∞𝑗!Γ(𝑗𝛼+𝛼+𝛽−2𝛼𝛽−2𝑚𝛼)_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝐽²¹𝜆_𝑛𝜙²(𝑟)𝑡^{∞−𝛼𝛽−𝑚𝛼+𝛼+𝛽−1}_𝑗=0(𝑗+𝑚+1)!−𝑑𝜆^2𝑛𝑡^𝛼/𝑎^𝑗+2_∞𝑗!Γ(𝑗𝛼+𝛼+𝛽−𝛼𝛽−𝑚𝛼)_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{∞𝛼−𝑚𝛼−1}_𝑗=0(𝑗+𝑚+1)!−𝑑𝜆^2𝑛𝑢^𝛼/𝑎^𝑗𝑢(𝑟,𝑡)=2_∞𝑗!Γ(𝛼𝑗+𝛼−𝑚𝛼)𝑓(𝑡−𝑢)𝑑𝑢.(3.16)_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝐽²¹𝜆_𝑛𝜙¹(𝑟)𝑡^{−2𝛼𝛽−𝑚𝛼+𝛼+𝛽−1}⋅𝐻^1,11,2𝑑𝜆^2𝑛𝑡^𝛼+2𝑎_∞𝑎||||(0,1),(1−(𝛼−−𝑚𝛽−+1,21𝛼)𝛽+2𝑚𝛼,𝛼)_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝐽²¹𝜆_𝑛𝜙²(𝑟)𝑡^{−𝛼𝛽−𝑚𝛼+𝛼+𝛽−1}⋅𝐻^1,11,2𝑑𝜆^2𝑛𝑡^𝛼+2_∞𝑎||||(0,1),(1−(−𝛼𝑚−−𝛽1+,1𝛼)𝛽+𝑚𝛼,𝛼)_∞𝑛=0_𝑚=0(−1)^𝑛𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{𝛼−𝑚𝛼−1}𝐻^1,11,2𝑑𝜆^2𝑛𝑢^𝛼𝑎||||(0,1)(,−(1𝑚−−𝛼1+,1𝑚)𝛼,𝛼)𝑓(𝑡−𝑢)𝑑𝑢.(3.17)

Example 3.2. Solve the differential equation (3.4) with initial condition 𝐼𝐼^{𝑡(1−𝛽)(1−2𝛼)}𝑢(𝑟,0)=0,_{𝑡(1−𝛽)(1−𝛼)}𝑢𝑢(𝑟𝑢(,𝑟(𝑡,𝑟)𝑡,)=𝑡)=0=fin0eivteefor𝑢yraw(t𝑟𝑟h,𝑟=0e=)r1e=,0f,o𝑡0r,𝑡>𝑡>0≤,00.,(3.18)

Solution 2. Taking Laplace and Hankel transform of (3.4), we get 𝑢̃̃(𝑟,𝑠)=𝐽¹𝜆_𝑛𝜆_𝑛𝑓(𝑠)𝑠^2𝛼+𝑎𝑠^𝛼+𝑑𝜆^2𝑛,(3.19) on taking Inverse Laplace transform of equation (3.19), we get 𝑢̃(𝑟,𝑡)=𝐿⁻¹𝑓(𝑠)𝐽¹𝜆_𝑛𝜆_𝑛𝐿⁻¹𝑠^2𝛼+𝑎𝑠1^𝛼+𝑑𝜆^2𝑛.(3.20)
By using convolution theorem for Laplace transform and taking inverse Hankel transform, we get 𝑢(𝑟,𝑡)=2^∞_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{𝛼−𝑚𝛼−1}𝐸^{𝑚𝛼,𝛼−2𝑚𝛼}−𝑑𝜆^2𝑛𝑢^𝛼𝑎𝑓(𝑡−𝑢)𝑑𝑢,(3.21) or 𝑢(𝑟,𝑡)=2^∞_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{∞𝛼−𝑚𝛼−1}_𝑗=0(𝑗+𝑚+1)!(𝑗)!−𝑑𝜆^2𝑛𝑢^𝛼/𝑎^𝑗Γ(𝛼𝑗+𝛼−𝑚𝛼).(3.22) By using the relation (2.2) 𝑢(𝑟,𝑡)=2^∞_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{𝛼−𝑚𝛼−1}𝐻^1,11,2𝑑𝜆^2𝑛𝑢^𝛼𝑎||||(0,1(−),𝑚(1−−1𝛼,1+)𝑚𝛼,𝛼)𝑓(𝑡−𝑢)𝑑𝑢,(3.23) or 𝑢(𝑟,𝑡)=2^∞_∞𝑛=0_𝑚=0(−1)^𝑚𝑎^𝑚+1𝐽₀𝜆_𝑛𝑟𝜆_𝑛𝐽₁𝜆_𝑛^𝑡0𝑢^{𝛼−𝑚𝛼−1}1Γ(𝑚)¹Ψ₁(𝛼(−𝑚2,1𝑚)𝛼;,𝛼);−𝑑𝜆^2𝑛𝑢^𝛼𝑎,(3.24) which is the required solution.