Analytical solutions for fractional partial delay differential-algebraic equations with Dirichlet boundary conditions defined on a finite domain

In this paper, we investigate the solution of multi-term time-space fractional partial delay differential-algebraic equations (MTS-FPDDAEs) with Dirichlet boundary conditions defined on a finite domain. We use Laplace transform method to give the solutions of multi-term time fractional delay differential-algebraic equations (MTS-FDDAEs). Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the MTS-FPDDAEs into the MTS-FDDAEs. By applying our obtained solutions to the resulting MTS-FDDAEs, the desired analytical solutions of the MTS-FPDDAEs are obtained. Finally, we give the solutions of some special cases.


Introduction
In the last few years, the interest of the scientific community towards fractional calculus experienced an exceptional boost, so that its applications can now be found in a great variety of science fields, for example, anomalous diffusion [1], solute transport [2], random and disordered media [3,4], electrical circuits [5], and so on. The reason of the success of the fractional calculus in modeling natural phenomena is that the fractional calculus operators are nonlocal operators, that makes them suitable to describe the long memory or nonlocal effects characterizing most of the physical phenomena.
Fractional partial differential equations are an important class of differential equations. They can model the dynamics of complex systems. At the same time, the necessity of a powerful technique for solving these new types of equations came up, becoming one of the main research objects in both the fields of theoretical and applied sciences, In the available literature, there exist various methods for solving fractional partial differential equations, such as analytical methods and numerical algorithms, [6][7][8][9][10].
Analytical solutions of fractional partial equations are of fundamental importance in describing and understanding the physical phenomena, since all the parameters are expressed in a mathematically closed form and therefore the influence of individual parameters on natural phenomena can be easily examined. Also, the analytical solutions make it easy to study asymptotic behaviors of the solutions, which are usually difficult to obtain through numerical calculations. Besides, the analytical solutions may serve as tools in assessing the computational performance and accuracy of numerical solutions.
The analytical solutions of fractional partial equations have been reported in literature, see e.g. [11][12][13][14][15][16][17][18][19][20][21][22][23]. Integral transforms, such as the Laplace transform and the Mellin transform, have been widely applied to develop the analytical solutions to fractional differential equations (for example, see [11,12]). But, one of the disadvantages of the integral transform method is that the inverse transform is mostly performed based on the complex functions, thus limiting the types of the analytical solutions of the equations. Accordingly, some researchers developed a spectral representation technique to obtain the desired solutions [20][21][22].
Fractional partial delay differential-algebraic equations often arise in many important physical problems, [24][25][26][27]. To the authors' knowledge, the analytical solutions of the fractional partial delay differential-algebraic equations have not been reported in literature yet. In [28], Zaczkiewicz applied Laplace transform for investigate of linear stationary fractional differential-algebraic equations with delays and to obtain analytical representation of solutions in the form of series in power of solutions to the equations. In this paper, we consider the following MTS-FPDDAEs defined on a finite domain of the form (1.1) with the nonhomogeneous Dirichlet boundary conditions u(0, t) = u 0 (t), u(L, t) = u L (t), 0 t T , (1.3) where (x, t) ∈ [0, L] × [0, T ] (L and T are constants), the operator P(D * t )u(x, t) is defined as u(x, t), 0 α p < · · · < α 1 < α 2, a i 0, and assume that 0 < α − α 1 1, D o t (o stands for α or α i ) is the Caputo fractional derivative of order o with respect to t, and the Laplacian operator is defined The space fractional derivative (− ) κ 2 (κ = p 1 , p 2 , q 1 , q 2 , and 1 < p 1 , q 2 2, 0 < q 1 , p 2 1) is a fractional Laplacian operator defined through the eigenfunction expansion on a finite domain. The detailed definitions of the Caputo fractional derivative and the fractional Laplacian operator are given in the next section (or see [20,29]).
The rest of this paper is organized as follows. In Section 2, we give some basic definitions and useful properties, which will be used in the paper. In Section 3, we use the Laplace transform to discuss the analytical solutions of fractional delay differential-algebraic equations. In Section 4, we discuss the analytical solutions of MTS-FPDDAEs with fractional diffusion terms. In Section 5, we give the analytical solutions of MTS-FPDDAEs with fractional wave terms. In Section 6, we provide the details of the solutions of some special cases.

Preliminaries
In this section, we give some basic definitions about fractional calculus and important properties on Laplace transform, which will be used throughout this paper. For details, one can refer to [29].
Definition 1 Let f ∈ C([0, T ]) and α > 0. The Riemann-Liouville fractional integral of order α with respect to t is defined as where Γ (·) is the Gamma function.
The Caputo fractional derivative of order α with respect to t is defined as The Laplace transform of the Caputo derivative of f is given by where F(s) denotes the Laplace transform of the function f (t).
There exists the following relationship between the Riemann-Liouville fractional integral and the Caputo fractional derivative. Property 1 ( [29]) Let m − 1 < α m, where m ∈ N + . Then the relations hold: Definition 3 ( [20]) Suppose that the Laplacian (− ) has a complete set of orthonormal eigenfunctions ϕ n corresponding to eigenvalues μ 2 n on a bounded region D, i.e., is one of the standard three homogeneous boundary conditions. Let then for any f ∈ F, (− ) α 2 is defined by Lemma 1 ( [20]) Suppose that the one-dimensional Laplacian (− ) defined with Dirichlet boundary conditions at x = 0 and x = L has a complete set of orthonormal eigenfunctions ϕ n corresponding to eigenvalues μ 2 n on a bounded region [0, L]. If (− )ϕ n = μ 2 n ϕ n on [0, L], and ϕ n (0) = ϕ n (L) = 0, then, the eigenvalues are given by μ 2 n = n 2 π 2 L 2 , and the corresponding eigenfunctions are ϕ n (x) = sin(nπ x/L), n = 1, 2, . . ..
An important function occurring in electrical systems is the delayed unit step function and its Laplace transform is given by and L{u a (t) f (t − a); s} = e −as F(s), a 0, (2.6) and also we have Another important function is the unit impulsive function and one knows that With respect to the convolution of the unit impulsive function, there exist several basic properties, which will be used in the latter discussion.

Property 2
Let δ be the unit impulsive function, f be any continuous function defined on D, and * denote the convolution operation. Then, the following statements hold: Finally, we introduce some notations, which will be used in the remain sections. We use C m n to denote the combinational number formula, * stands for the convolution operation, and r t := t h , where t h denotes the largest integer less than or equal to t h .

Solution representation
In this section, we investigate the solution representations of fractional differentialalgebraic equations with delays. Consider a simple fractional differential-algebraic equation with delay of the form: where λ 1 , λ 2 , λ 3 , λ 4 ∈ R, and u, e are two known continuous functions defined on First, we consider the solution of equation (3.1) in the case 0 < α 1. When we discuss its solution, equation (3.1) satisfies the initial conditions and we assume that the initial conditions are consistent, i.e., ϕ(0) = λ 3 x 1 (0) + λ 4 ϕ(−h) + e(0). For giving the solution of equation (3.1) with the initial conditions (3.2), we take the Laplace transforms on the both sides of equation (3.1) to obtain where X 1 (S), U (s) and E(s) denote the Laplace transforms of x 1 (t), u(t) and e(t), respectively. Next, we consider the Laplace inverse transform of X 1 (s).
For brevity, we introduce the following two notations: Then, by (2.2) and (2.7), we have Using the similar arguments to (3.5), we can obtain and and On the other hand, we define a new staircase function and extend the function Based on this extension, it has the following relationship (3.10) Therefore, from (3.3), (3.4), (3.5), (3.6), (3.7), (3.8) and (3.10) and the definition (3.9) of the function p(t), we can obtain that: For 0 t < h, the representation of the solution x 1 is given by For h t T , we have On the other hand, we solve the second equation of (3.1) by step-wise method to deduce that Based on the above analysis, we can establish the following theorem.
In the following, we consider the solutions of equation (3.1) in the case 1 < α 2. We discuss its solution that satisfies the initial conditions (3.13) and assume that ϕ satisfies In this case, using the similar arguments as in Theorem 1, we can establish the following theorem.
. Then the solutions of equation (3.1) with the initial conditions (3.13) are given by where λ and e ρ;λz α,β are defined in (3.4).
At this stage, we consider the solution of MTS-FPDDAE of the form: (3.15) where the operator P(D * t ) is defined as First, we consider the solutions of equation (3.15) in the case 0 α p < · · · < α 1 < α 1. When we discuss its solution, the equation satisfies the initial conditions and we assume that ϕ satisfies To provide the solution of (3.15) with the initial conditions (3.16), we need some useful lemmas. (3.4), and p i=2 a i s α i +λ s α +a 1 s α 1 < 1. Then we have where k i 0, i = 1, 2, . . . , p.

Remark 1
The series involved in Lemma 2 converges uniformly on [0, T ]. That is to say, the function Φ(t) is well-defined. In fact, from the relationship 0 < α − α 1 1 and one knows that the three-parameter Mittag-Leffler functions involved in Lemma 2 are monotone decaying functions of t and so they are largest at t = 0. (For details, one can refer to [30]) That is to say, we have Obviously, the series of the right hand of inequality (3.17) is convergent uniformly on [0, T ]. Hence, the series involved in Lemma 2 is convergent uniformly on [0, T ].
Using the similar arguments, we can prove that the series involved in Lemmas 3 and 4 converge uniformly on [0, T ]. Therefore, the function Ψ (t) and Υ (t) are also well-defined.
Now we can give the analytical representation of the solution.
Proof Using the similar proof to Theorem 1, we take the Laplace transforms on the both sides of equation (3.15) to obtain Next, we consider the Laplace inverse transform of X 1 (s). Let Then, we can deduce that (3.20) On the one hand, by Lemma 3 and (2.7), one knows that where Ψ is defined in Lemma 3. On the other hand, since where δ is the unit impulsive function, and Φ is defined in Lemma 2, it follows that (3.23) Based on the above analysis and Property 2, we can obtain Similarly, we can obtain that and

From (3.24), (3.25), (3.26) and (3.27), the Laplace inverse transform x 1 (t) of X 1 (s)
can be obtained. Furthermore, using the stepwise method, we can give the expression of x 2 (t). The proof is completed.
Using similar arguments to Theorem 3, we can establish the following theorem.
Proof We firstly take the Laplace transforms on the both sides of equation (3.15) to obtain Using the similar arguments to (3.24), we have And we combine (3.25), (3.26) and (3.27) to obtain the expression of x 1 (t). The proof is completed.

Analytical solutions for the MTS-FPDDAES with fractional diffusion terms
In this section, we consider the analytical solutions of the MTS-FPDDAE in the case 0 α s < · · · < α 1 < α 1 and a 1 > 0. In this case, equation (1.1) is a generalized time and space fractional partial delay differential-algebraic equation with fractional diffusion terms. We discuss its solution that satisfies the nonhomogeneous Dirichlet boundary condition (1.2) and the initial conditions In order to solve the equation with the nonhomogeneous Dirichlet boundary conditions, we first transform the nonhomogeneous Dirichlet boundary conditions into homogeneous Dirichlet boundary conditions. Let where W 1 (x, t), W 2 (x, t) are two new unknown functions, and Substituting (4.2) into (1.1), we get the fractional partial delay differential-algebraic equation with the homogeneous Dirichlet boundary conditions and the initial conditions According to Lemma 1, the eigenvalues μ 2 n (n = 1, 2, . . .) of the operator (− ) with the homogeneous boundary conditions is μ 2 n = n 2 π 2 /L 2 , and the corresponding eigenfunctions are ϕ n (x) = sin(nπ x/L), n = 1, 2, . . .. Then we set (4.7) Substituting (4.6) and (4.7) into (4.3) leads to (4.8) with the initial conditions where −h t 0. By Theorem 3, the solutions of equation (4.8) with the initial conditions (4.9) are and (4.11) where and Φ, Ψ are defined in Lemmas 2 and 3, respectively. Therefore, we obtain the solutions of equation (1.1) with the boundary condition (1.2) and the initial conditions (3.11) are where w n1 (t), w n2 (t), λ i (i = 1, 2, 3, 4) and λ are given in (4.10), (4.11) and (4.12), respectively.

Analytical solutions of the MTS-FPDDAES with fractional wave terms
In this section, we consider the analytical solutions of the MTS-FPDDAE in the case 0 α p < · · · < α p 0 1 < α p 0 +1 < · · · < α 2 and a 1 > 0. In this case, equation (1.1) is a generalized time and space fractional partial delay differential-algebraic equation with fractional wave terms. We discuss its solution that satisfies boundary condition (1.2) and the initial conditions In order to solve the equation with the nonhomogeneous Dirichlet boundary conditions, we firstly transform the nonhomogeneous Dirichlet boundary conditions into Dirichlet homogeneous boundary conditions. Let where W 1 (x, t), W 2 (x, t) are two new unknown functions, and Substituting (5.2) into (1.1), we get the following fractional differential equation with the homogeneous Dirichlet boundary conditions and the initial conditions According to Lemma 1, the eigenvalues μ 2 n (n = 1, 2, . . .) of the operator (− ) with the homogeneous boundary conditions is μ 2 n = n 2 π 2 /L 2 , and the corresponding eigenfunctions are ϕ n (x) = sin(nπ x/L), n = 1, 2, . . .. Then we set  (5.9) with the initial conditions By Theorem 4, the solutions of (5.9) with the initial conditions (5.10), (5.11) and (5.12) are (5.13) and (5.14) where λ i (i = 1, 2, 3, 4) are defined as (4.12), and Φ, Ψ and Υ are defined in Lemmas 2, 3 and 4, respectively. Therefore, we obtain the solutions of Eq. (1.1) with the boundary condition (1.2) and the initial condition (3.11) are where w n1 (t) and w n2 (t) are given in (5.13) and (5.14), respectively.

Special cases
In this section, we provide details of solutions for some special cases.