Exact Solution of a Non-Homogeneous Fractional Differential Equation with a Variable Coefficient and Its Applications

Abstract

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function.

1. Introduction

Fractional differential equations have become an important topic in applied mathematics due to their generalization of classical models and their ability to describe memory and hereditary effects, as well as phenomena in porous media [1]. For applications of fractional calculus in bioengineering, viscoelasticity, control theory, image processing, and economic dynamics, we refer the reader to [2,3,4,5]. Several book series on fractional calculus also provide systematic information on its wide range of applications. In particular, Volumes 4 and 5 of the Handbook of Fractional Calculus with Applications [6,7] are devoted to applications in physics.

Special functions such as Mittag–Leffler functions, Prabhakar functions, hypergeometric functions, Meijer’s G-functions, and Fox’s H-functions play an important role in fractional calculus [8,9,10,11]. They arise in the solution of fractional differential equations [1,12] and have many applications in various areas. In particular, Meijer’s G-functions appear in statistical distributions, functional equations, theoretical physics, hydrodynamics, sinusoidal signals, and optimization problems [13]. Furthermore, special functions are often used as kernels in generalized operators of integration and differentiation [14,15]. For instance, in [15], Kirykova and Al-Saqabi studied a hyper-Bessel integral equation with a kernel involving Meijer G-functions. They solved the integral equation using the transmutation method, and the obtained solutions are represented in terms of series expansions involving Meijer G-functions.

In this paper, we study the following non-homogeneous fractional differential equation with a variable coefficient:

$$t^{-\beta }{}^{C}D^{\alpha }y\left(t\right)+\lambda y\left(t\right)=f\left(t\right),0<\alpha <1,\beta \in \mathbb{R},\lambda \ne 0.$$

(1)

Some relevant works have considered the corresponding homogeneous fractional differential equation. Various methods have been employed to derive explicit solutions, which are typically expressed in terms of special functions. For example, in [1], the authors expressed the solution of the homogeneous fractional differential equation in terms of a generalized Mittag–Leffler type function using the method of reduction to Volterra integral equation. In addition, Klimek [16] studied homogeneous fractional differential equations but with derivatives appearing on the right-hand side. The obtained solutions were represented in the form of series involving Meijer G-functions and were derived using the Mellin transform method.

Subsequently, Klimek and Dziembowski [17] considered a special case of the non-homogeneous fractional differential Equation (1), where the non-homogeneous term is given by Meijer’s G-functions. The general solution of the considered equation is obtained as a sum of Meijer’s G-functions series. The convergence of the series is analyzed using theorems related to Meijer’s G-functions (see the references therein for further details).

More general fractional differential equations with variable coefficients are studied by Arran et al. in [18]. These equations involve fractional-order derivatives of either Riemann–Liouville or Caputo types, with variable coefficients that can be either integrable or continuous functions. The authors derived explicit solutions in the form of convergent infinite series involving compositions of fractional integrals. To establish the uniqueness of the solutions, they applied the Banach fixed point theorem in suitable function spaces.

A Cauchy problem including a related fractional differential equation with a variable coefficient was investigated in [19], where the authors derived an explicit representation of the solution in terms of a generalized Mittag–Leffler-type function and an integral kernel represented by a recursively defined series expansion. This result was later generalized to the case of the Hilfer fractional derivative in [20]. We also mention the works [21,22], in which decay estimates and initial-value problems for time-fractional evolution equations with time-dependent coefficients were studied. In addition, exact solutions of various fractional-order differential equations were investigated in [23,24,25,26,27,28].

Partial differential equations with variable coefficients may be regarded as degenerate partial differential equations [29]. Homogeneous time-fractional partial differential equations with degeneration have been investigated in several works [30,31,32,33]. In particular, Turmetov and Kadirkulov studied a mixed-type equation involving a Caputo fractional derivative and a degenerate spatial operator in [30]. Another mixed-type equation with three lines of degeneration was investigated in [31]. In [32], the authors considered a mixed-type problem for higher-order equations involving fractional derivatives and degeneration in both variables. More recently, an inverse problem with a Bitsadze–Samarskii-type condition for a time-fractional partial differential equation was studied in [33]. Existence results for fractional problems involving singular non-linearities and non-local convective terms were recently obtained by Gambera and Marano [34]. Although their work concerns the fractional $(p,q)$-Laplacian with Dirichlet boundary conditions, some of the employed techniques, such as the sub–supersolution method combined with truncation arguments, may also be useful in the study of exact solutions or particular classes of solutions for fractional partial differential equations with variable coefficients. Inverse-initial problems for time-degenerate fractional partial differential equations involving bi-ordinal Hilfer derivatives were recently studied by Karimov et al. [35]. By applying the method of separation of variables, the authors showed that the corresponding solutions depend essentially on the solution of a Cauchy problem for a homogeneous fractional differential equation with a variable coefficient involving bi-ordinal Hilfer derivatives in the time variable. An explicit solution of this Cauchy problem was derived in terms of the Kilbas–Saigo function.

Moreover, recent studies on fractional differential equations with variable coefficients have mainly focused on the qualitative analysis of solutions. For instance, Li et al. [36] investigated a non-linear fractional partial integro-differential equation involving a Caputo fractional derivative and multi-term integral operators subject to non-local boundary conditions. The authors established existence and uniqueness results by applying Banach’s fixed-point theorem together with Babenko’s method. In another direction, Rahmonov et al. [37] studied an inverse problem for a two-dimensional fractional wave equation with a variable coefficient. By reducing the problem to an equivalent integral equation, they proved the existence, uniqueness, and stability of the solution as well as the unknown coefficient.

To the best of our knowledge, although the literature on fractional differential equations with variable coefficients has grown considerably in recent years, explicit general solutions of non-homogeneous problems in terms of special functions have received comparatively limited attention. In this paper, we aim to contribute to this area by deriving an analytical solution of a non-homogeneous fractional differential equation with a variable coefficient and by demonstrating the important role of special functions, such as generalized Mittag–Leffler functions and Meijer G-functions, in fractional calculus. Explicit analytical representations are useful for qualitative analysis, asymptotic investigations, and numerical verification. Moreover, the obtained general solution can be employed in the study of a class of variable-coefficient fractional differential equations arising in real-world applications.

The rest of this paper is organized as follows. In the next section, some preliminaries related to Meijer’s G-functions are given. In Section 3, the solution to the non-homogeneous fractional differential Equation (1) is derived. Section 4 is devoted to some examples to illustrate the main result. In the last three sections, we present direct and inverse source problems involving fractional differential equations with variable coefficients along with example solutions.

2. Preliminaries

In this section, we recall some preliminaries which are needed in this paper. First, we recall the definition of Meijer’s G-functions $G_{p,q}^{m,n}$.

Definition 1

 ([14]). Let $m,n,p,q\in N_0,$ $0\le m\le q,$ $0\le n\le p$ and let $a_i,b_i\in \mathbb{C}$. The Meijer’s G-function $G_{m,n}^{p,q}\left(z\right)$ is given in terms of a Mellin–Barnes-type integral

$$\begin{array}{cc}{G}_{p,q}^{m,n}\left(z\right)=\hfill & G_{p,q}^{m,n}\left(z|\begin{array}{c}{a}_1,\cdots ,a_p\hfill \\ b_1,\cdots ,b_q\hfill \end{array}\right)=G_{p,q}^{m,n}\left(z|\begin{array}{c}{\left(a_j\right)}_1^p\hfill \\ {\left(b_k\right)}_1^q\hfill \end{array}\right)\hfill \\ & {\displaystyle ={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathcal{L}}{\displaystyle \frac{{\displaystyle {\displaystyle \prod _{k=1}^m}\Gamma (b_k-s){\displaystyle \prod _{j=1}^n}\Gamma (1-a_j+s)}}{{\displaystyle {\displaystyle \prod _{k=m+1}^q}\Gamma (a_j-s){\displaystyle \prod _{j=n+1}^p}\Gamma (1-b_k+s)}}}{z}^{s}ds,}\hfill \end{array}$$

where $\mathcal{L}$ is a suitably chosen contour. For more details about this definition, see [13,14].

Also, we need the following integral of Meijer’s G-function:

Lemma 1

 ([14]). If $Re{a}_k>Re{b}_k>-1,$$k=1,\cdots ,m,$ then the following integral formula holds:

$${\int }_0^1{G}_{m,m}^{m,0}\left(x|\begin{array}{c}{\left(a_k\right)}_1^m\hfill \\ {\left(b_k\right)}_1^m\hfill \end{array}\right)dx={\displaystyle \prod _{k=1}^m}{\displaystyle \frac{\Gamma (b_k+1)}{\Gamma (a_k+1)}}.$$

(2)

Since Meijer’s G-functions are extensions of the generalized hypergeometric functions [14], we need to define some particular cases of hypergeometric functions like Gauss’s hypergeometric function and Appell’s two-variable hypergeometric function [38]. The integral representation of Gauss’s hypergeometric function ${}_{2}F_1(a,b;c;x)$ is defined as

$$\begin{array}{cc}\hfill & {}_{2}F_1(a,b;c;x)={\displaystyle \frac{\Gamma \left(c\right)}{\Gamma \left(b\right)\Gamma (c-b)}}{\int }_0^1{t}^{b-1}{(1-t)}^{c-b-1}{(1-tx)}^{-a}dt,\hfill \\ & a\in \mathbb{R},c>b>0\hfill \end{array}$$

(3)

and its series representation is given by

$${}_{2}F_1(a,b;c;x)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}}{\displaystyle \frac{x^n}{n!}},\left|x\right|<1,$$

(4)

where ${\left(a\right)}_n$ stands for Pochammer’s symbol, which has the following properties:

$${\left(a\right)}_n={\displaystyle \frac{\Gamma (a+n)}{\Gamma \left(a\right)}}\mathrm{and}{\left(a\right)}_{n+k}={\left(a\right)}_k{(a+k)}_n.$$

(5)

Moreover, Appell’s two-variable hypergeometric function $F_3$ is defined by double series representation as follows:

$$F_3(a,a^{\prime },b,b^{\prime };c;x,y)={\displaystyle \sum _{m,n=0}^{\infty }}{\displaystyle \frac{{\left(a\right)}_m{\left(a^{\prime }\right)}_n{\left(b\right)}_m{\left(b^{\prime }\right)}_n{x}^m{y}^n}{{\left(c\right)}_{m+n}m!n!}},(max(\left|x\right|,\left|y\right|<1\left)\right).$$

(6)

where $c\ne 0,-1,-2,\dots$. Now, we give some known explicit expressions of $G_{m,m}^{m,0}$ in terms of hypergeometric functions [14,38].

First, let ${\displaystyle {\psi }_m={\displaystyle \sum _{j=1}^m}(b_j-a_j),}$ then

• For $m=1:$ the G-function is expressed in terms of power functions

  $$G_{1,1}^{1,0}\left(z|\begin{array}{c}{b}_1\hfill \\ a_1\hfill \end{array}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{z^{a_1}{(1-z)}^{{\psi }_1-1}}{\Gamma \left({\psi }_1\right)}},\hfill & \mathrm{for}0<z<1,\hfill \\ 0,\hfill & \mathrm{for}z>1\hfill \end{array}\right.$$

  and for $a_1=0$ and $b_1=\alpha$, we have

  $$G_{1,1}^{1,0}\left(z|\begin{array}{c}\alpha \hfill \\ 0\hfill \end{array}\right)={\displaystyle \frac{{(1-z)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}.$$

  (7)
  One can also find the following general form:

  $$\begin{array}{cc}\hfill & G_{m,m}^{m,0}\left(z|\begin{array}{c}{\left(j\alpha \right)}_1^m\hfill \\ {\left((j-1)\alpha \right)}_1^m\hfill \end{array}\right)=G_{m,m}^{m,0}\left(z|\begin{array}{c}\alpha ,2\alpha ,\cdots m\alpha \hfill \\ 0,\alpha ,\cdots ,(m-1)\alpha \hfill \end{array}\right)=\hfill \\ \hfill & ={\displaystyle \frac{{(1-z)}^{m\alpha -1}}{\Gamma \left(\alpha \right)}}.\hfill \end{array}$$

  (8)
• For $m=2$: the G-function is expressed in terms of the Gauss hypergeometric function

  $$\begin{array}{cc}& G_{2,2}^{2,0}\left(z|\begin{array}{c}{b}_1,b_2\hfill \\ a_1,a_2\hfill \end{array}\right)=\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \frac{z^{a_2}{(1-z)}^{{\psi }_2-1}}{\Gamma \left({\psi }_2\right)}}{}_{2}F_1(b_1-a_1,b_2-a_1;{\psi }_2;1-z),\hfill & \mathrm{for}z<1,\hfill \\ 0,\hfill & \mathrm{for}z>1.\hfill \end{array}\right.\hfill \end{array}$$
• For $m=3$: the G-function can be represented in terms of the Appell hypergeometric function as follows:

  $$\begin{array}{cc}& G_{3,3}^{3,0}\left(z|\begin{array}{c}{b}_1,b_2,b_3\hfill \\ a_1,a_2,a_3\hfill \end{array}\right)={\displaystyle \frac{z^{a_1+a_2-b_1}{(1-z)}^{{\psi }_3-1}}{\Gamma \left({\psi }_3\right)}}\times \hfill \\ & \times F_3(b_1-a_2,b_3-a_3,b_1-a_1,b_2-a_3;{\psi }_3;1-{\displaystyle \frac{1}{z}},1-z).\hfill \end{array}$$

  For more properties of Meijer’s G-functions, the reader is referred to [13,14,39,40]. Now, we recall the definition of the multiple Erdelyi–Kober operator:

Definition 2

 ([14]). Let $m\ge 1$ be an integer, $\eta >0,$${\gamma }_1,\dots ,{\gamma }_m$ and ${\delta }_1,\dots ,{\delta }_m>0$ be arbitrary real numbers. Consider set $\gamma =({\gamma }_1,\cdots ,{\gamma }_m)$ as a multi-weight, $\delta =({\delta }_1,\cdots ,{\delta }_m)$ as a positive multi-order of integration. For functions $f\in C_{\alpha },$ $\alpha \ge \underset{k}{max}\left[-\eta ({\gamma }_k+1)\right]$, we define the multiple Erdelyi–Kober operators in the following way:

$$I_{\eta ,m}^{\left({\gamma }_k\right),\left({\delta }_k\right)}f\left(x\right)={\int }_0^1{G}_{m,m}^{m,0}\left(t|\begin{array}{c}{({\gamma }_k+{\delta }_k)}_1^m\hfill \\ {\left({\gamma }_k\right)}_1^m\hfill \end{array}\right)f(t^{{\displaystyle \frac{1}{\eta }}}x)dt,$$

(9)

where $C_{\alpha }:=\left\{f\left(x\right)=x^p\stackrel{\backsim }{f}\left(x\right);p>\alpha ,\stackrel{\backsim }{f}\in C[0,\infty )\right\}.$

In the following lemmas, we present the multiple Erdelyi–Kober operator of the power function and the G-function:

Lemma 2

 ([14]). The multiple Erdelyi–Kober operators (9) are well defined in the space $C_{\alpha }$ with $\alpha \ge {\displaystyle \underset{k}{\max }}\left[-\eta ({\gamma }_k+1)\right]$ and preserve there (up to a constant multiplier) the power functions:

$$I_{\eta ,m}^{\left({\gamma }_k\right),\left({\delta }_k\right)}{x}^p=C_p{x}^p,$$

(10)

where ${\displaystyle C_p={\displaystyle \prod _{k=1}^m}{\displaystyle \frac{\Gamma ({\gamma }_k+{\displaystyle \frac{p}{\eta }}+1)}{\Gamma ({\gamma }_k+{\delta }_k+{\displaystyle \frac{p}{\eta }}+1)}}}$ and $p>\alpha .$

Lemma 3

 ([14]). The $I_{\eta ,m}^{\left({\gamma }_k\right),\left({\delta }_k\right)}$ image of a G-function of $C_{\alpha }$ is also a G-function whose last three components of the order $(p,n;q,r)$ are increased by m, namely

$$I_{\eta ,m}^{\left({\gamma }_k\right),\left({\delta }_k\right)}\left\{G_{q,r}^{p,n}\left(x^{\eta }w|\begin{array}{c}{\left(a_i\right)}_1^q\hfill \\ {\left(b_j\right)}_1^r\hfill \end{array}\right)\right\}=G_{q,r+m}^{p+m,n+m}\left(x^{\eta }w|\begin{array}{c}{(-{\gamma }_k)}_1^m,{\left(a_i\right)}_1^q\hfill \\ {\left(b_j\right)}_1^r,{(-{\gamma }_k-{\delta }_k)}_1^m\hfill \end{array}\right).$$

(11)

Finally, we recall the Riemann–Liouville fractional derivative of order $0<\alpha <1$,

$${\displaystyle D^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_0^t\frac{f\left(\tau \right)}{{(\tau -t)}^{\alpha }}d\tau ,}$$

and Caputo fractional derivative of order $0<\alpha <1$ is [1]

$${}^{C}D^{\alpha }f\left(t\right)=D^{\alpha }(f\left(t\right)-f\left(0\right)).$$

The following property is also needed when solving the fractional differential equation under consideration [1]:

$$I^{\alpha }{D}^{\alpha }f\left(t\right)=f\left(t\right),0<\alpha <1,$$

where

$${\displaystyle I^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t\frac{f\left(\tau \right)}{{(t-\tau )}^{1-\alpha }}d\tau ,t>0,}$$

is the Riemann–Liouville fractional integral.

3. Main Result

Consider the following initial value problem with a non-homogeneous fractional differential equation:

$$t^{-\beta } {}^{C}D^{\alpha }y\left(t\right)+\lambda y\left(t\right)=f\left(t\right),y\left(0\right)=y_0,0<\alpha <1,\beta \ge 0,\lambda \ne 0,t>0,$$

with $f\in C_{\alpha }$. Here, we look for a continuous solution $y\left(t\right)$ that satisfies the above problem by reducing it to an integral equation and using the successive approximation method.

First, writing the Caputo derivative in terms of a Riemann–Liouville derivative, the above problem can be written as

$$D^{\alpha }(y\left(t\right)-y_0)+\lambda t^{\beta }y\left(t\right)=t^{\beta }f\left(t\right).$$

(12)

Then, applying the Riemann–Liouville fractional integral to both sides of (12), we obtain the following integral equation:

$$y\left(t\right)=y_0-\lambda I^{\alpha }{t}^{\beta }y\left(t\right)+I^{\alpha }{t}^{\beta }f\left(t\right).$$

Now, using the successive approximation method to solve the above integral equation, starting with $y_0=y\left(0\right)$ and

$$y_n\left(t\right)=y_0-\lambda I^{\alpha }{t}^{\beta }{y}_{n-1}\left(t\right)+I^{\alpha }{t}^{\beta }f\left(t\right),$$

we get

$$\begin{array}{cc}{y}_1\left(t\right)\hfill & =y_0-\lambda I^{\alpha }{t}^{\beta }{y}_0+I^{\alpha }{t}^{\beta }f\left(t\right)\hfill \\ & =y\left(0\right)\left(1-\lambda {\displaystyle \frac{\Gamma (\beta +1)}{\Gamma (\alpha +\beta +1)}}{t}^{\alpha +\beta }\right)+I^{\alpha }{t}^{\beta }f\left(t\right).\hfill \end{array}$$

Next, we find $y_2\left(t\right)$ as follows:

$$\begin{array}{c}{y}_2\left(t\right)=y_0-\lambda I^{\alpha }{t}^{\beta }{y}_1\left(t\right)+I^{\alpha }{t}^{\beta }f\left(t\right)\hfill \\ =y\left(0\right)-\lambda I^{\alpha }{t}^{\beta }\left(y\left(0\right)-\lambda y\left(0\right){\displaystyle \frac{\Gamma (\beta +1)}{\Gamma (\alpha +\beta +1)}}{t}^{\alpha +\beta }+I^{\alpha }{t}^{\beta }f\left(t\right)\right)+I^{\alpha }{t}^{\beta }f\left(t\right)\hfill \\ =y\left(0\right)\left(1-\lambda {\displaystyle \frac{\Gamma (\beta +1)}{\Gamma (\alpha +\beta +1)}}{t}^{\alpha +\beta }+{\lambda }^2{\displaystyle \frac{\Gamma (\beta +1)\Gamma (2\beta +\alpha +1)}{\Gamma (\alpha +\beta +1)\Gamma (2\alpha +2\beta +1)}}{t}^{2(\alpha +\beta )}\right)\hfill \\ +I^{\alpha }{t}^{\beta }f\left(t\right)-\lambda I^{\alpha }{t}^{\beta }{I}^{\alpha }{t}^{\beta }f\left(t\right)\hfill \\ =y\left(0\right)\left({\displaystyle 1+\sum _{k=1}^2{(-\lambda )}^k\prod _{r=0}^{k-1}{\displaystyle \frac{\Gamma (\alpha r+\beta (r+1)+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(r+1)+1)}}{t}^{k(\alpha +\beta )}}\right)\hfill \\ +I^{\alpha }{t}^{\beta }f\left(t\right)-\lambda I^{\alpha }{t}^{\beta }{I}^{\alpha }{t}^{\beta }f\left(t\right).\hfill \end{array}$$

Similarly, $y_3\left(t\right)$ can be obtained as

$$\begin{array}{cc}{y}_3\left(t\right)\hfill & =y\left(0\right)\left({\displaystyle 1+\sum _{k=1}^3{(-\lambda )}^k\prod _{r=0}^{k-1}{\displaystyle \frac{\Gamma (\alpha r+\beta (r+1)+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(r+1)+1)}}{t}^{k(\alpha +\beta )}}\right)+I^{\alpha }{t}^{\beta }f\left(t\right)\hfill \\ & -\lambda I^{\alpha }{t}^{\beta }{I}^{\alpha }{t}^{\beta }f\left(t\right)+{\lambda }^2{I}^{\alpha }{t}^{\beta }{I}^{\alpha }{t}^{\beta }{I}^{\alpha }{t}^{\beta }f\left(t\right).\hfill \end{array}$$

Hence, the nth term can be written as

$$y_n=y_{hn}+y_{pn},$$

where

$$y_{hn}=y\left(0\right)\left({\displaystyle 1+\sum _{k=1}^n{(-\lambda )}^k\prod _{r=0}^{k-1}{\displaystyle \frac{\Gamma (\alpha r+\beta (r+1)+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(r+1)+1)}}{t}^{k(\alpha +\beta )}}\right)$$

and

$${\displaystyle y_{pn}={\displaystyle \sum _{m=1}^n}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^{1}f\left(xt\right)G_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx,}$$

For detailed derivation of the above closed form of $y_{pn}$ as a series of integrals of G-functions, see Appendix A.

Hence, as $n\to \infty$, the general solution of the fractional differential Equation (1) can be represented in terms of a generalized Mittag–Leffler-type function and a Meijer G-function as follows:

$$\begin{array}{ccc}\hfill y\left(t\right)& =& y\left(0\right)E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })\hfill \\ & +& {\displaystyle {\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^{1}f\left(xt\right)G_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx,}\hfill \end{array}$$

where the generalized Mittag–Leffler-type function, $E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })$, is defined as

$$E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })={\displaystyle \sum _{k=0}^{\infty }}{C}_k{(-\lambda t^{\alpha +\beta })}^k,$$

${\displaystyle C_0=1,C_k={\displaystyle \prod _{r=0}^{k-1}}{\displaystyle \frac{\Gamma (\alpha r+\beta (r+1)+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(r+1)+1)}}.}$

Now, we show the absolute convergence of the series in the above equation. First, let

$$\begin{array}{cc}{a}_m\hfill & :={(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}\left|I_{1,m}^{\left({\gamma }_j\right),\left({\delta }_j\right)}f\left(t\right)\right|\hfill \end{array}$$

Then, for a fixed time $t,$ $f\in C_{\alpha }$ and using the Lemma 1, we have

$$\begin{array}{c}|I_{1,m}^{\left({\gamma }_j\right),\left({\delta }_j\right)}f\left(t\right)|\le A{\int }_0^1\left|G_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)\right|dx\hfill \\ =A{\displaystyle {\prod }_{k=1}^m}\left|{\displaystyle \frac{\Gamma \left(k\right(\beta +\alpha )-\alpha +1)}{\Gamma \left(k\right(\beta +\alpha )+1)}}\right|,0<\alpha <1,\beta \ge 0,\hfill \end{array}$$

where A is a positive constant. Using the above, we have

$$\left|{\displaystyle \frac{a_{m+1}}{a_m}}\right|\le A{\displaystyle \frac{{\prod }_{k=1}^{m+1}\left|{\displaystyle \frac{\Gamma \left(k\right(\beta +\alpha )-\alpha +1)}{\Gamma \left(k\right(\beta +\alpha )+1)}}\right|}{{\prod }_{k=1}^m\left|{\displaystyle \frac{\Gamma \left(k\right(\beta +\alpha )-\alpha +1)}{\Gamma \left(k\right(\beta +\alpha )+1)}}\right|}}=A\left|{\displaystyle \frac{\Gamma \left(\right(m+1\left)\right(\beta +\alpha )-\alpha +1)}{\Gamma \left(\right(m+1\left)\right(\beta +\alpha )+1)}}\right|$$

Hence, using Stirling’s approximation, we prove that $\underset{m\to \infty }{lim}\left|{\displaystyle \frac{a_{m+1}}{a_m}}\right|=0$, which implies that the series is absolutely convergent. This establishes the pointwise convergence of the solution. Moreover, the result can be extended to uniform convergence on any compact subinterval of $(0,\infty )$.

The main result can be summarized in the following theorem:

Theorem 1.

Consider the following non-homogeneous fractional differential equation

$$t^{-\beta } {}^{C}D_{0t}y\left(t\right)+\lambda y\left(t\right)=f\left(t\right),0<\alpha <1,\beta \ge 0,\lambda \ne 0,t>0,$$

with $f\in C_{\alpha }$ and a constant initial condition $y\left(0\right)=y_0$. Then, it has the following general solution

$$\begin{array}{c}y\left(t\right)=y\left(0\right)E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })\hfill \\ {\displaystyle +{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^{1}f\left(tx\right)G_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx,}\hfill \\ {\displaystyle =y\left(0\right)E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })+{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{I}_{1,m}^{\left({\gamma }_j\right),\left(\delta \right)}f\left(t\right)}\hfill \end{array}$$

(13)

where ${\gamma }_j=j(\alpha +\beta )-\alpha ,$ ${\delta }_j=\alpha$ and

$$y\left(t\right)=y\left(0\right)E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })$$

is the solution of the corresponding homogeneous differential equation.

4. Example Solutions of the Main Result

In this section, we present several examples to illustrate the main result for certain choices of the non-homogeneous term $f\left(t\right)$ and selected model parameters. The functions $f\left(t\right)$ are chosen either from the literature to demonstrate that previously obtained special cases can be derived from the general solution or for mathematical convenience in order to evaluate the integrals involving Meijer G-functions explicitly.

Example 1.

Consider $f\left(t\right)=f_0$, then using (2), the particular solution is given by

$$\begin{array}{c}{\displaystyle y_p\left(t\right)=f_0{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^1{G}_{m,m}^{m,0}\left(u|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx}\hfill \\ {\displaystyle =f_0{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{\displaystyle \prod _{j=1}^m}{\displaystyle \frac{\Gamma (j\beta +(j-1)\alpha +1)}{\Gamma (j\beta +j\alpha +1)}}{t}^{m(\alpha +\beta )}}\hfill \\ {\displaystyle =f_0{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{\Gamma \left(\right(k+1)\beta +k\alpha +1)}{\Gamma \left(\right(k+1)\beta +(k+1)\alpha +1)}}{t}^{m(\alpha +\beta )}}\hfill \\ {\displaystyle =-{\displaystyle \frac{f_0}{\lambda }}{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^m{C}_m{t}^{m(\alpha +\beta )}={\displaystyle \frac{f_0}{\lambda }}\left({\displaystyle 1-E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })}\right),}\hfill \end{array}$$

where ${\displaystyle C_m={\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{\Gamma (\alpha k+\beta (k+1)+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(k+1)+1)}}.}$

Now, the general solution can be written as

$$y\left(t\right)=\left(y\left(0\right)-{\displaystyle \frac{f_0}{\lambda }}\right)E_{\alpha ,1+\beta /\alpha ,\beta /\alpha }(-\lambda t^{\alpha +\beta })+{\displaystyle \frac{f_0}{\lambda }}.$$

Example 2.

Let $f\left(t\right)=t^n,n\in \mathbb{R}$ then using (10), the particular solution is given by

$$\begin{array}{c}{\displaystyle y_p\left(t\right)={\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^1{\left(xt\right)}^n{G}_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx}\hfill \\ {\displaystyle ={\displaystyle \sum _{m=1}^{\infty }}{C}_m{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )+n}}\hfill \\ {\displaystyle =-{\displaystyle \frac{t^n}{\lambda }}{\displaystyle \sum _{m=1}^{\infty }}{C}_m{(-\lambda )}^m{t}^{m(\alpha +\beta )}}\hfill \\ ={\displaystyle \frac{t^n}{\lambda }}\left(1-E_{\alpha ,1+\beta /\alpha ,(\beta +n)/\alpha }(-\lambda t^{\alpha +\beta })\right),\hfill \end{array}$$

where ${\displaystyle C_m={\displaystyle \prod _{k=0}^{m-1}}{\displaystyle \frac{\Gamma (\alpha k+\beta (k+1)+n+1)}{\Gamma \left(\right(\alpha +\beta \left)\right(k+1)+n+1)}}.}$ Note that for $n=0$, the solution coincides with the one obtained in the previous example for the constant case.

Example 3.

For $f\left(t\right)=G_{q,r}^{p,n}\left(t|\begin{array}{c}{\left(a_i\right)}_1^q\hfill \\ {\left(b_l\right)}_1^r\hfill \end{array}\right).$

Then, using (11), the particular solution is reduced to

$${\displaystyle \begin{array}{c}\begin{array}{c}{y}_p\left(t\right)=\hfill \end{array}{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}\hfill \\ \times {\int }_0^1{G}_{q,r}^{p,n}\left(xt|\begin{array}{c}{\left(a_i\right)}_1^q\hfill \\ {\left(b_l\right)}_1^r\hfill \end{array}\right)G_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)dx\hfill \\ ={\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{G}_{q+m,r+m}^{p,n+m}\left(t|\begin{array}{c}{(-j\beta -(j-1)\alpha )}_1^m,{\left(a_i\right)}_1^q\hfill \\ {\left(b_l\right)}_1^r,{(-j\beta -j\alpha )}_1^m,\hfill \end{array}\right).\hfill \end{array}}$$

The above solution is the same as the solution found in [17] for this particular choice.

Example 4.

Let $f\left(t\right)=sin\omega \sqrt{t},$$\omega >0$, which can be written as a G-function:

$$sin\left(\omega \sqrt{t}\right)=\sqrt{\pi }{G}_{0,2}^{1,0}\left({\displaystyle \frac{t{\omega }^2}{4}}|\begin{array}{cc}& \\ {\displaystyle \frac{1}{2}},\hfill & 0\hfill \end{array}\right).$$

Now, the particular solution is given by

$$\begin{array}{c}{\displaystyle y_p\left(t\right)=\sqrt{\pi }{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}}\hfill \\ {\displaystyle \times {\int }_0^1{G}_{m,m}^{m,0}\left(x|\begin{array}{c}{(j\beta +j\alpha )}_1^m\hfill \\ {(j\beta +(j-1)\alpha )}_1^m\hfill \end{array}\right)G_{0,2}^{1,0}\left(\frac{x{\omega }^2}{4}|\begin{array}{cc}& \\ {\displaystyle \frac{1}{2}},\hfill & 0\hfill \end{array}\right)dx}\hfill \\ {\displaystyle =\sqrt{\pi }{\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{m(\alpha +\beta )}{G}_{m,m+2}^{1,m}\left({\displaystyle \frac{{\omega }^2}{4}}|\begin{array}{cc}& {(-j\beta -(j-1)\alpha )}_1^m\hfill \\ 0,{\displaystyle \frac{1}{2}},\hfill & {(-j\beta -j\alpha )}_1^m\hfill \end{array}\right.}\hfill \end{array}$$

Example 5.

For $\beta =0,$ Equation (1) reduces to

$${}^{C}D^{\alpha }y\left(t\right)+\lambda y\left(t\right)=f\left(t\right),y\left(0\right)=y_0,0<\alpha <1,\beta \ge 0,\lambda \ne 0.$$

and the homogeneous solution part of (13) becomes

$$y_h\left(t\right)=y\left(0\right)E_{\alpha ,1,0}(-\lambda t^{\alpha })=y\left(0\right)E_{\alpha ,1}(-\lambda t^{\alpha }).$$

The particular solution reduces to

$$\begin{array}{cc}\hfill y_p\left(t\right)& {\displaystyle ={\displaystyle \sum _{m=1}^{\infty }}{(-\lambda )}^{m-1}{t}^{\alpha m}{\int }_0^{1}f\left(tx\right)G_{m,m}^{m,0}\left(x|\begin{array}{c}{\left(i\alpha \right)}_1^m\hfill \\ {\left((i-1)\alpha \right)}_1^m\hfill \end{array}\right)dx}\hfill \\ & {\displaystyle =t^{\alpha }{\int }_0^{1}f\left(tx\right)G_{1,1}^{1,0}\left(x|\begin{array}{c}\alpha \hfill \\ 0\hfill \end{array}\right)dx-\lambda t^{2\alpha }{\int }_0^{1}f\left(tx\right)G_{2,2}^{2,0}\left(x|\begin{array}{c}\alpha ,2\alpha \hfill \\ 0,\alpha \hfill \end{array}\right)dx}\hfill \\ & {\displaystyle +{\lambda }^2{t}^{3\alpha }{\int }_0^{3}f\left(tx\right)G_{3,3}^{3,0}\left(x|\begin{array}{c}\alpha ,2\alpha ,3\alpha \hfill \\ 0,\alpha ,2\alpha \hfill \end{array}\right)dx-\cdots .}\hfill \end{array}$$

Using (7), we get

$$\begin{array}{cc}\hfill y_p\left(t\right)& {\displaystyle ={\displaystyle \frac{t^{\alpha }}{\Gamma \left(\alpha \right)}}{\int }_0^{1}f\left(tx\right){(1-x)}^{\alpha -1}dx-{\displaystyle \frac{\lambda t^{2\alpha }}{\Gamma \left(2\alpha \right)}}{\int }_0^{1}f\left(tx\right){(1-x)}^{2\alpha -1}dx}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\lambda }^2{t}^{3\alpha }}{\Gamma \left(3\alpha \right)}}{\int }_0^{1}f\left(tx\right){(1-x)}^{3\alpha -1}dx-\cdots .}\hfill \end{array}$$

Putting $u=xt$, we obtain

$$\begin{array}{cc}\hfill y_p\left(t\right)& {\displaystyle ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{t}f\left(u\right){(t-u)}^{\alpha -1}du-{\displaystyle \frac{\lambda }{\Gamma \left(2\alpha \right)}}{\int }_0^{t}f\left(u\right){(t-u)}^{2\alpha -1}du}\hfill \\ & {\displaystyle +{\displaystyle \frac{{\lambda }^2}{\Gamma \left(3\alpha \right)}}{\int }_0^{t}f\left(u\right){(t-u)}^{3\alpha -1}du-\cdots .}\hfill \\ & {\displaystyle ={\int }_0^{t}f\left(u\right){\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(-\lambda )}^{k-1}{(t-u)}^{\alpha k-1}}{\Gamma \left(\alpha k\right)}}du}\hfill \\ & {\displaystyle ={\int }_0^{t}f\left(u\right){(t-u)}^{\alpha -1}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{(-\lambda )}^k{(t-u)}^{\alpha k}}{\Gamma (\alpha k+\alpha )}}du}\hfill \\ & {\displaystyle ={\int }_0^{t}f\left(u\right){(t-u)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-\lambda {(t-u)}^{\alpha }\right)du.}\hfill \end{array}$$

The obtained solution coincides with the well-known solution of the above equation [1].

In the remaining three sections, we demonstrate the importance of the considered fractional differential Equation (1) in solving direct and inverse source problems involving fractional differential equations with variable coefficients. Such problems arise in a variety of practical applications, including contaminant transport in heterogeneous aquifers [41,42], anomalous diffusion in biological tissues related to drug delivery [2,43], and heat conduction in materials with memory such as viscoelastic solids [3,44]. For example, in contaminant transport through heterogeneous aquifers, pollutants such as heavy metals or radionuclides diffuse through fractured or porous media. Classical Fickian diffusion models become inadequate because the pore-size distribution and trapping effects may cause particles to remain immobilized for random waiting times, which naturally leads to fractional-order models. The unknown source term $f\left(x\right)$ may represent a buried leaking tank or landfill whose spatial profile must be reconstructed from concentration measurements taken at two different times. This constitutes a classical inverse source identification problem in hydrogeology. If the source function is known, then the problem reduces to a standard direct problem.

5. An Inverse Source Problem

Find a regular solution of a pair of functions $\left\{u\right(x,t),f(x\left)\right\}$ in the domain $\Omega =\left\{0<x<L,0<t<T\right\}$ that satisfies the following equation

$$t^{\alpha -1}{}^{C}D_{0|t}^{\alpha }u(x,t)-k{u}_{xx}(x,t)=f\left(x\right),(x,t)\in \Omega ,0<\alpha <1,$$

(14)

the Neumann boundary conditions

$$u_x(0,t)=0,u_x(L,t)=0,0\le t\le T,$$

(15)

and the initial and final time conditions

$$u(x,0)=\varphi \left(x\right),u(x,T)=\psi \left(x\right)0\le x\le L,$$

(16)

where $\varphi \left(x\right)$ and $\psi \left(x\right)$ are given functions, k is a positive constant.

We begin by solving the homogeneous equation corresponding to (14) along with the boundary conditions (15) using the separation of variables. Thus, we have the following self-adjoint problem:

$$X^{″}+\lambda X=0,X^{\prime }\left(0\right)=0,X^{\prime }\left(L\right)=0.$$

(17)

The above problem has the following set of eigenvalues and eigenfunctions:

$${\lambda }_n={\left({\displaystyle \frac{n\pi }{L}}\right)}^2\mathrm{and}{X}_n=cos\left({\displaystyle \frac{n\pi x}{L}}\right),n=0,1,2,\cdots$$

(18)

Thus, the solutions $u(x,t)$ and $f\left(x\right)$ can be written in series expansion forms as

$$u(x,t)=u_0\left(t\right)+{\displaystyle \sum _{n=1}^{\infty }}{u}_n\left(t\right)cos\left({\displaystyle \frac{n\pi x}{L}}\right),$$

(19)

and

$$f\left(x\right)=f_0+{\displaystyle \sum _{n=1}^{\infty }}{f}_{n}cos\left({\displaystyle \frac{n\pi x}{L}}\right),$$

(20)

where $u_0\left(t\right),u_n\left(t\right),f_0$, and $f_n$ are the unknowns to be found. Substituting (19) and (20) into (14) and (16), we obtain the following problems involving a linear fractional differential with variable coefficients:

$$t^{\alpha -1}{}^{C}D_{0|t}^{\alpha }{u}_0\left(t\right)=f_0,u_0\left(0\right)={\varphi }_0,u_0\left(T\right)={\psi }_0,$$

(21)

and

$$t^{\alpha -1}{}^{C}D_{0|t}^{\alpha }{u}_n\left(t\right)+k{\lambda }_n{u}_n\left(t\right)=f_n,u_n\left(0\right)={\varphi }_n,u_n\left(T\right)={\psi }_n$$

(22)

where, ${\varphi }_0,$${\psi }_0$ ${\varphi }_n$, and ${\psi }_n$ are the coefficients of the series expansions of $\varphi \left(x\right)$ and $\psi \left(x\right)$ in terms of the orthogonal basis (18), i.e.,

$$\begin{array}{cc}{\varphi }_0={\displaystyle \frac{1}{L}}{\int }_0^L\varphi \left(x\right)dx,\hfill & {\psi }_0={\displaystyle \frac{1}{L}}{\int }_0^L\psi \left(x\right)dx,\hfill \\ {\varphi }_n={\displaystyle \frac{2}{L}}{\int }_0^L\varphi \left(x\right)cos\left({\displaystyle \frac{n\pi x}{L}}\right)dx,\hfill & {\psi }_n={\displaystyle \frac{2}{L}}{\int }_0^L\psi \left(x\right)cos\left({\displaystyle \frac{n\pi x}{L}}\right)dx.\hfill \end{array}$$

The solution of problem (21) is given by

$$u_0\left(t\right)={\varphi }_0+{\displaystyle \frac{({\psi }_0-{\varphi }_0)t}{T}}\mathrm{and}{f}_0={\displaystyle \frac{{\psi }_0-{\varphi }_0}{\Gamma (2-\alpha )T}},$$

and the solution of problem (22) is given by

$$u_n\left(t\right)={\varphi }_n-C_n\left(1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\lambda }_{n}kt\right)\right)\mathrm{and}{f}_n={\lambda }_{n}k({\varphi }_n-C_n),$$

where

$$C_n={\displaystyle \frac{{\varphi }_n-{\psi }_n}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\lambda }_{n}kT\right)}}.$$

Hence, the expression of $u(x,t)$ and $f\left(x\right)$ can be written as

$$\begin{array}{cc}u(x,t)=\hfill & {\displaystyle {\varphi }_0+{\displaystyle \frac{({\psi }_0-{\varphi }_0)t}{T}}-{\displaystyle \sum _{n=1}^{\infty }}{C}_n\left(1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\lambda }_{n}kt\right)\right)cos\left(\frac{n\pi x}{L}\right)}\hfill \\ & {\displaystyle +{\displaystyle \sum _{n=1}^{\infty }}{\varphi }_{n}cos\left(\frac{n\pi x}{L}\right)}\hfill \\ & =({\psi }_0-{\varphi }_0){\displaystyle \frac{t}{T}}+\varphi \left(x\right)\hfill \\ & {\displaystyle -{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kt\right)}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kT\right)}}({\varphi }_n-{\psi }_n)cos\left(\frac{n\pi x}{L}\right)}\hfill \end{array}$$

and

$$\begin{array}{cc}f\left(x\right)\hfill & {\displaystyle ={\displaystyle \frac{{\psi }_0-{\varphi }_0}{\Gamma (2-\alpha )T}}+{\displaystyle \sum _{n=1}^{\infty }}{\lambda }_{n}k({\varphi }_n-C_n)cos\left(\frac{n\pi x}{L}\right)}\hfill \\ & {\displaystyle ={\displaystyle \frac{{\psi }_0-{\varphi }_0}{\Gamma (2-\alpha )T}}-{\varphi }^{″}\left(x\right)-{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left(\frac{n\pi }{L}\right)}^{2}k({\varphi }_n-{\psi }_n)}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kT\right)}}cos\left(\frac{n\pi x}{L}\right).}\hfill \end{array}$$

To complete the proof of the existence of the solution, we need to show the uniform convergence of the series representations of $u(x,t),$ $u_{xx}(x,t),$${}^{C}D_{0|t}^{\alpha }u(x,t)$, and $f\left(x\right).$

We first estimate the generalized Mittag–Leffler-type function using the result of Theorem 2 in [45]:

$${\displaystyle \frac{1}{1+\Gamma (1-\alpha ){\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kt}}\le E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kt\right)\le {\displaystyle \frac{1}{1+\Gamma (2-\alpha ){\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kt}}$$

so the expression of $u(x,t)$ can be estimated as follows:

$$\begin{array}{c}{\displaystyle \left|u(x,t)\right|\le \left|\varphi \left(x\right)\right|+\left(\right|{\psi }_0|+|{\varphi }_0\left|\right)+{\displaystyle \sum _{n=1}^{\infty }}\left(\right|{\psi }_n|+|{\varphi }_n\left|\right){\displaystyle \frac{\Gamma (1-\alpha ){\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kT}{1+\Gamma (1-\alpha ){\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kT}}}\hfill \\ {\displaystyle \le \left|\varphi \left(x\right)\right|+\left(\right|{\psi }_0|+|{\varphi }_0\left|\right)+c_T{\displaystyle \sum _{n=1}^{\infty }}\left(\right|{\psi }_n|+|{\varphi }_n\left|\right).}\hfill \end{array}$$

Using standard Fourier coefficients estimates through integration by parts, the inequality $2ab\le a^2+b^2$, and the Bessel’s inequality, we have

$${\displaystyle \left|u(x,t)\right|\le \left|\varphi \left(x\right)\right|+\left(\right|{\psi }_0|+|{\varphi }_0\left|\right)+{\displaystyle \frac{c_T}{2}}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{L^2}{{\left(n\pi \right)}^2}}+{\displaystyle \frac{c_T}{2}}{\parallel {\psi }^{\prime }\parallel }_{L^2(0,L)}^2+{\displaystyle \frac{c_T}{2}}{\parallel {\varphi }^{\prime }\parallel }_{L^2(0,L)}^2}$$

where $c_T=1+{\displaystyle \frac{L^2}{kT{\pi }^2\Gamma (2-\alpha )}}$ and

$$\begin{array}{c}{\displaystyle {\varphi }_{1n}={\displaystyle \frac{2}{L}}{\int }_0^L{\varphi }^{\prime }\left(x\right)sin\left(\frac{n\pi x}{L}\right)dx,\mathrm{and}{\psi }_{1n}={\displaystyle \frac{2}{L}}{\int }_0^L{\psi }^{\prime }\left(x\right)sin\left(\frac{n\pi x}{L}\right)dx.}\hfill \end{array}$$

Assuming that $\varphi \left(x\right)\in C[0,L],$${\psi }^{\prime }\left(x\right),{\varphi }^{\prime }\left(x\right)\in L^2(0,L)$, then, by Weierstrass M-test, the series representation of $u(x,t)$ is uniformly convergent. Similarly, using higher-order Fourier coefficient estimates, we have the following estimate for $f\left(x\right)$:

$$\begin{array}{c}{\displaystyle \left|f\left(x\right)\right|\le |{\varphi }^{″}\left(x\right)|+{\displaystyle \frac{1}{\Gamma (2-\alpha )T}}\left(|{\varphi }_0|+|{\psi }_0|\right)+k{c}_T{\displaystyle \sum _{n=1}^{\infty }}{\left({\displaystyle \frac{n\pi }{L}}\right)}^2\left(|{\varphi }_n|+|{\psi }_n|\right)}\hfill \\ \le |{\varphi }^{″}\left(x\right)|+{\displaystyle \frac{1}{\Gamma (2-\alpha )T}}\left(|{\varphi }_0|+|{\psi }_0|\right)+{\displaystyle \frac{kL}{2\pi }}{c}_T\left({\displaystyle \sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^2}}+{\parallel {\psi }^{‴}\parallel }_{L^2(0,L)}^2+{\parallel {\varphi }^{‴}\parallel }_{L^2(0,L)}^2}\right)\hfill \end{array}$$

where

$$\begin{array}{c}{\displaystyle {\varphi }_{3n}={\displaystyle \frac{2}{L}}{\int }_0^L{\varphi }^{‴}\left(x\right)sin\left(\frac{n\pi x}{L}\right)dx,\mathrm{and}{\psi }_{3n}={\displaystyle \frac{2}{L}}{\int }_0^L{\psi }^{‴}\left(x\right)sin\left(\frac{n\pi x}{L}\right)dx.}\hfill \end{array}$$

Assuming ${\varphi }^{\prime }\left(0\right)={\varphi }^{\prime }\left(L\right)=0,$ ${\psi }^{\prime }\left(0\right)={\psi }^{\prime }\left(L\right)=0,$ ${\psi }^{\prime \prime \prime }\left(x\right),{\varphi }^{\prime \prime \prime }\left(x\right)\in L^2(0,L)$ and using Weierstrass M-test leads to the uniform convergence of the series representation of $f\left(x\right)$.

One can proceed similarly to show the uniform convergence of the series representation of ${}^{C}D_{0|t}^{\alpha }u(x,t)$ and $u_{xx}(x,t)$.

The main result for this section can be summarized in the following theorem:

Theorem 2.

Assume that $\varphi \left(x\right),\psi \left(x\right)\in C^2[0,L],$ such that ${\varphi }^{\prime }\left(0\right)={\psi }^{\prime }\left(0\right)=0,$ ${\varphi }^{\prime }\left(L\right)={\psi }^{\prime }\left(L\right)=0,$ ${\psi }^{‴}\left(x\right),{\varphi }^{‴}\left(x\right)\in L^2(0,L),$ then the inverse source problem (14)–(16) has a unique pair of solutions $\left\{u(x,t),f\left(x\right)\right\}$ given by

$$\begin{array}{cc}u(x,t)\hfill & =({\psi }_0-{\varphi }_0){\displaystyle \frac{t}{T}}+\varphi \left(x\right)\hfill \\ & {\displaystyle -{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kt\right)}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kT\right)}}({\varphi }_n-{\psi }_n)cos\left(\frac{n\pi x}{L}\right)}\hfill \end{array}$$

(23)

and

$$\begin{array}{c}{\displaystyle f\left(x\right)={\displaystyle \frac{{\psi }_0-{\varphi }_0}{\Gamma (1-\alpha )T}}-{\varphi }^{″}\left(x\right)-{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\left(\frac{n\pi }{L}\right)}^{2}k({\varphi }_n-{\psi }_n)}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left(\frac{n\pi }{L}\right)}^{2}kT\right)}}cos\left(\frac{n\pi x}{L}\right),}\hfill \end{array}$$

(24)

where,

$$\begin{array}{cc}{\varphi }_0={\displaystyle \frac{1}{L}}{\int }_0^L\varphi \left(x\right)dx,\hfill & {\psi }_0={\displaystyle \frac{1}{L}}{\int }_0^L\psi \left(x\right)dx,\hfill \\ {\varphi }_n={\displaystyle \frac{2}{L}}{\int }_0^L\varphi \left(x\right)cos\left({\displaystyle \frac{n\pi x}{L}}\right)dx,\hfill & {\psi }_n={\displaystyle \frac{2}{L}}{\int }_0^L\psi \left(x\right)cos\left({\displaystyle \frac{n\pi x}{L}}\right)dx.\hfill \end{array}$$

6. A Direct Problem

Find a function $u(x,t)$ in a domain Ω such that $u(·,t)\in C^2[0,L]$ and $u(x,·)\in C[0,T]$ satisfying

$$t^{\alpha -1}{}^{C}D_{0|t}^{\alpha }u(x,t)-k{u}_{xx}(x,t)=f(x,t),(x,t)\in \Omega ,$$

(25)

the Dirichlet boundary conditions

$$u(0,t)=0,u(L,t)=0,0\le t\le T,$$

(26)

and the initial condition

$$u(x,0)=\varphi \left(x\right),0\le x\le L,$$

(27)

where $f(x,t)$ is a given source function, $k>0$, $0<\alpha <1.$ Solving the homogeneous part of Equation (25) using separation of variables and using the boundary condition (26), we obtain the following spectral problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{X}^{″}+\lambda X=0,\hfill \\ X\left(0\right)=0,X\left(L\right)=0.\hfill \end{array}\right.\end{array}$$

(28)

It is known that the above problem is self-adjoint and has the following eigenvalues and eigenfunctions:

$${\lambda }_n={\left({\displaystyle \frac{n\pi }{L}}\right)}^2,X_n=sin\left({\displaystyle \frac{n\pi x}{L}}\right),n=1,2,3,\cdots$$

(29)

Using the fact that the system of eigenfunctions in (29) forms an orthogonal basis in $L^2(0,L)$ [46], we can write the solution $u(x,t)$ and $f(x,t)$ in the form of a series expansion as follows:

$$u(x,t)={\displaystyle \sum _{n=1}^{\infty }}{u}_n\left(t\right)sin\left({\displaystyle \frac{n\pi x}{L}}\right),$$

(30)

and

$$f(x,t)={\displaystyle \sum _{n=1}^{\infty }}{f}_n\left(t\right)sin\left({\displaystyle \frac{n\pi x}{L}}\right),$$

(31)

where $u_n\left(t\right)$ is the unknown to be determined and the coefficient $f_n\left(t\right)$ is known and it is given by

$$f_n\left(t\right)={\displaystyle \frac{2}{L}}{\int }_0^{L}f(x,t)sin\left({\displaystyle \frac{n\pi x}{L}}\right)dx.$$

Substituting (30) and (31) into (25) and (27), we get the linear fractional differential equation

$$t^{\alpha -1}{}^{C}D_{0|t}^{\alpha }{u}_n\left(t\right)+k{\lambda }_n{u}_n\left(t\right)=f_n\left(t\right),$$

(32)

with the following condition

$$u_n\left(0\right)={\varphi }_n,$$

(33)

where ${\varphi }_n$ is the coefficient of the series expansion of $\varphi \left(x\right)$ in terms of the orthogonal basis (29), i.e.,

$${\varphi }_n={\displaystyle \frac{2}{L}}{\int }_0^L\varphi \left(x\right)sin\left({\displaystyle \frac{n\pi x}{L}}\right)dx.$$

Whereupon using Theorem 1, the solution of Equation (32) is given by

$$\begin{array}{cc}{u}_n\left(t\right)=\hfill & u_n\left(0\right)E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\lambda }_{n}kt\right)+\hfill \\ & {\displaystyle {\displaystyle \sum _{m=1}^{\infty }}{(-{\lambda }_{n}k)}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^1{f}_n\left(st\right)G_{m,m}^{m,0}\left(s|\begin{array}{c}{\left(j\right)}_1^m\hfill \\ {(j-\alpha )}_1^m\hfill \end{array}\right)ds.}\hfill \end{array}$$

(34)

Therefore, the series solution $u(x,t)$ can be written as

$$\begin{array}{c}{\displaystyle u(x,t)={\displaystyle \sum _{n=1}^{\infty }}{\Phi }_n{E}_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}kt\right)sin\left({\displaystyle \frac{n\pi x}{L}}\right)+{\displaystyle \sum _{n=1}^{\infty }}sin\left({\displaystyle \frac{n\pi x}{L}}\right)}\hfill \\ {\displaystyle \times \left({\displaystyle \sum _{m=1}^{\infty }{\left(-{\left({\displaystyle \frac{n\pi }{L}}\right)}^{2}k\right)}^{m-1}{t}^{m(\alpha +\beta )}{\int }_0^1{f}_n\left(st\right)G_{m,m}^{m,0}\left(s|\begin{array}{c}{\left(j\right)}_1^m\hfill \\ {(j-\alpha )}_1^m\hfill \end{array}\right)ds}\right)}\hfill \end{array}$$

(35)

Appropriate conditions are needed on the generalized Mittag–Leffler-type $E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{\lambda }_{n}t\right)$, Meijer’s G-functions $G_{m,m}^{m,0}$ and the given functions $f(x,t)$ and $\varphi \left(x\right)$ for establishing the uniform convergence of the series expansions of $u(x,t)$, $u_{xx}(x,t),$ ${}^{C}D_{0|t}^{\alpha }u(x,t)$.

7. Illustrative Examples for the Inverse and Direct Problems

Here, we provide example solutions illustrating the general results of the inverse and direct problems for specific choices of the initial-time data, final-time data, and source function.

Example 6.

For the inverse problem, we choose $L=\pi ,k=1,\psi \left(x\right)=0,$ and $\varphi \left(x\right)=cos\left(x\right)$. Thus, using (23) and (24), we have

$$u(x,t)=cosx-{\displaystyle \frac{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-t\right)}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-T\right)}}cosx$$

and

$$f\left(x\right)=cosx-{\displaystyle \frac{cosx}{1-E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-T\right)}}.$$

Example 7.

For the direct problem, consider the following choices: $\varphi \left(x\right)=sinx,$$L=\pi ,k=1$, and $f(x,t)=k_0+ϵsin\omega \sqrt{t},\omega >0,ϵ>0.$

Then, using (35), the expression $u(x,t)$ can be written as

$$\begin{array}{c}u(x,t)=E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-t\right)sinx\hfill \\ {\displaystyle +{\displaystyle \frac{2k_0}{\pi }}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{(1-{(-1)}^n)}{n^3}}\left(E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-n^{2}t\right)-1\right)sinnx+{\displaystyle \frac{2ϵ}{\sqrt{\pi }}}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{(1-{(-1)}^n)}{n}}}\hfill \\ {\displaystyle \times \left({\displaystyle \sum _{m=1}^{\infty }}{(-n^2)}^{m-1}{t}^m{G}_{m,m+2}^{1,m}\left({\displaystyle \frac{{\omega }^{2}t}{4}}|\begin{array}{c}{(-j+\alpha )}_1^m\hfill \\ 0,{\displaystyle \frac{1}{2}},{(-j)}_1^m\hfill \end{array}\right)ds\right)sinnx.}\hfill \end{array}$$

In particular, when $ϵ=0$, we have

$$\begin{array}{cc}u(x,t)\hfill & =E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-t\right)sinx\hfill \\ & {\displaystyle +{\displaystyle \frac{4k_0}{\pi }}{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{{(2n-1)}^3}}\left(E_{\alpha ,1/\alpha ,1/\alpha -1}\left(-{(2n-1)}^{2}t\right)-1\right)sin(2n-1)x.}\hfill \end{array}$$

8. Conclusions

In this work, we studied a class of non-homogeneous fractional differential equations involving a Caputo fractional derivative and a variable coefficient. By reformulating the problem as an equivalent integral equation and applying the method of successive approximations, we obtained a general solution expressed in terms of a generalized Mittag–Leffler function and a series of integrals involving Meijer G-functions. The absolute convergence of this series was established using Stirling’s approximation. To demonstrate the effectiveness of the proposed approach, example solutions corresponding to specific non-homogeneous terms and selected model parameters were presented. These examples show that several previously obtained special cases from the literature can be derived from the general solution obtained in the present study. Furthermore, we investigated both direct and inverse source problems as applications of the main result. The solutions were constructed through series expansions using orthogonal bases derived via the method of separation of variables. Uniform convergence of the obtained series under certain conditions was proven for the inverse source problem. Finally, illustrative examples of the direct and inverse problems corresponding to specific choices of the initial-time data, final-time data, and source function were presented.