Inverse source problem for multi-term time-fractional diffusion equation with nonlocal boundary conditions

1.   Introduction

For a fixed positive integer $m$, let $\alpha_i$ and $q_i, \ (i = 1, ..., m)$ be positive constants such that $1 > \alpha_1 > \cdot\cdot\cdot > \alpha_m > 0$. We assume $q_1 = 1$ without loss of generality. Consider the inverse problem of finding a pair of functions $\{r, u\}$ such that it satisfies the equation

the initial condition

the nonlocal boundary conditions

and the overdetermination condition

where $T > 0; \ D_T = \{(x, t): \ 0 < x < 1, \ 0 < t\leq T\}$; $f, \ \varphi$ and $E$ are given functions; $a_0, \ a_1$ and $b_1$ are real constants such that $|a_1|+|b_1| > 0, \ a_1+b_1\neq0$; $\partial^{\alpha_i}_{t}$ is the Caputo time-fractional derivative defined in ^[1] by

and $\Gamma(\cdot)$ is the Gamma function. Note that, in the case $|a_1|+|b_1| > 0, \ a_1+b_1 = 0$, problem (1.1)–(1.4) will immediately be incorrect.

For (1.1)–(1.3), the direct problem is the determination of $u$ in $\overline{D}_T$ such that $u(\cdot, t)\in C^2[0, 1]$ and $\partial^{\alpha_i}_{t}{u(x, \cdot)}\in C(0, T]$ when the initial temperature $\varphi$ and the source term $rf$ are given and continuous.

If the function $r(t), \ t \in [0, T]$ is unknown, the inverse problem is formulated as the problem of finding a pair of functions $\{r, u\}$ which satisfy (1.1)–(1.4) with $r\in C[0, T], \ u(\cdot, t)\in C^2[0, 1], \ \partial^{\alpha_i}_{t}{u(x, \cdot)}\in C(0, T]$.

The integral condition (1.4) arises when the data on the boundary cannot be measured directly, but only the average value of the solution can be measured along the boundary ^[2].

Fractional calculus is used in quantum mechanics ^[3], biophysics ^[4], control theory ^[5], viscoelasticity ^[6], signal processing ^[7], biological sciences ^[8], and many other disciplines.

In ^[9], the inverse problem for the classical heat equation satisfying the boundary conditions

was considered. The boundary conditions (1.3) are the particular case of the boundary conditions that considered in ^[9] when $b_0 = 0, \ c_0 = d_0 = 1$ and $c_1 = d_1 = 0$.

The practical applications of nonlocal boundary value problems span across diverse fields, encompassing chemical diffusion ^[10], thermal conductivity ^[11], biological processes ^[12], and others. Particularly in scenarios like multiphase flows involving liquids, solids, and gases, heat flow is often proportional to the variations in boundary temperatures among distinct phases, alongside the parameters $a_0, a_1, b_1$ outlined in the nonlocal boundary conditions (1.3). This elucidates the growing importance and extensive utilization of inverse problems featuring nonlocal boundary conditions across various disciplines.

We note several papers devoted to the study the inverse problem for a time-fractional diffusion equation with nonlocal boundary conditions. In ^[13], the inverse source problem for the time-fractional diffusion equation in two dimensions was considered. ^[14] focused on determining a time-dependent factor of an unknown source under certain sub-boundary conditions for the time-fractional diffusion equation using nonlocal measurement data. The paper established the existence and uniqueness of the solution to the inverse source problem by applying Lax-Milgram's lemma in appropriate Sobolev spaces. In ^[15], the inverse source problem for the time-fractional diffusion equation in two dimensional space was considered, where the time fractional derivative is the Hilfer derivative. In ^[16], the authors considered the problem of determining the distribution and the source term for the time-fractional diffusion equation. Two inverse problems for the time-fractional diffusion equation with a family of nonlocal boundary conditions were discussed in ^[17].

We also note some papers related to the study of impulsive differential equations and the inverse source problem for a time-fractional diffusion equation. In ^[18], the study focused on multipoint BVPs concerning a generalized class of impulsive fractional-order nonlinear differential equations. ^[19] addressed the study of a class of impulsive integro-differential equations, where impulses are not instantaneous. In ^[20], the estimation of an unknown source term in the time-fractional diffusion equation from measurement data was explored using the alternating direction method of multipliers. The work presented in ^[21] involved the mathematical analysis of an inverse source problem governed by a time-fractional diffusion equation. The objectives of this research included identifying the source function using additional data via a regularized optimal control approach and determining the regularization parameters through bi-level optimization.

This paper is an extension of the problem considered in ^[22].

It is well known that the boundary conditions (1.3) are not strongly regular boundary conditions ^[23]. For this reason, when we solve the problem (1.1)–(1.4) by the Fourier method, then the system of eigenfunctions of the auxiliary spectral problem does not form a basis. Therefore, the application of the Fourier method is impossible and additional research is needed.

2.   Preliminaries

The multinomial Mittag–Leffler function is defined as ^[24]

where $0 < \beta_0 < 2, \ 0 < \beta_i < 1 \ (i = 1, ..., m)$, and $z_i\in \mathbb{C} \ (i = 1, ..., m)$. Here, $\left(k; k_1, ..., k_m\right)$ denotes the multinomial coefficient

and $k_i \ (1\leq i \leq m)$ are non-negative integers.

Lemma 2.1. ^[25] Let $0 < \beta_0 < 2, \ 0 < \beta_i < 1 \ (i = 1, ..., m)$, and $z_i\in \mathbb{C} \ (i = 1, ..., m)$ be fixed. Then,

Lemma 2.2. ^[25] Let $0 < \beta < 2$ and $1 > \alpha_1 > \cdot\cdot\cdot > \alpha_m > 0$ be given. Assume that $\alpha_1\pi/2 < \mu < \alpha_1\pi, \ \mu\leq |arg(z_1)|\leq \pi$, and there exists $K > 0$ such that $-K\leq z_i < 0, \ (i = 2, ..., m)$. Then, there exists a constant $c > 0$ depending on $\mu, \ K, \ \alpha_i \ (i = 1, ..., m)$, and $\beta$ such that

Let us denote

where $l_1, ..., l_m$ are some positive constants.

Lemma 2.3. ^[25] Let $1 > \alpha_1 > \cdot\cdot\cdot > \alpha_m > 0$. Then,

Lemma 2.4. ^[26] Let $f_n$ be a sequence of functions defined on $(a, b]$ for each $n\in \mathbf{N}$, such that:

(1). For some $\alpha > 0$, $\partial^{\alpha}_{t}{f_n(t)}$ exists for all $n\in \mathbf{N}$, $t\in (a, b]$;

(2). Both the series $\sum_{n = 1}^{\infty}{f_n}(t)$ and $\sum_{n = 1}^{\infty}{\partial^{\alpha}_{t}f_n}(t)$ are uniformly convergent on the interval $[a+\varepsilon, b]$ for any $\varepsilon > 0$.

Then, $\sum_{n = 1}^{\infty}{f_n}(t)$ is $\alpha > 0$ differentiable, where $\sum_{n = 1}^{\infty}{f_n}(t)$ is a series of functions that must satisfy

Consider the Volterra integral equation

Denote $\Delta: = \{(t, t_1): 0\leq t_1 < t\leq 1\}$ and introduce the class $S^{\alpha}$ of kernels $Q(t, t_1)$ that are defined and continuous on $\Delta$ and for $(t, t_1)\in \Delta$ satisfy the inequality

Lemma 2.5. ^[27] Let $g_1\in C[0, 1]$ and $Q(t, t_1)\in S^{\alpha}$ with $0\leq\alpha < 1$. Then, (2.1) has a unique solution $g_0\in C[0, 1]$.

3.   Functional relations

The application of the Fourier method to solve problem (1.1)–(1.4) leads to a spectral problem

where $|a_1|+|b_1| > 0, \ a_1+b_1\neq0$. The system of eigenfunctions of problem (3.1) does not form a basis in $L_2(0, 1)$ ^[28]. For this reason, we cannot solve problem (1.1)–(1.4) by the Fourier method. Therefore, using the method from ^[29], we introduce even $u_1$ and odd $u_2$ with respect to the variable $x$ parts of the function $u$ as

where

At the same time, for all $(x, t)\in D_T$, the relations

hold. Equality (3.2) implies boundary relations

Substituting $u_1$ and $u_2$ into boundary conditions (1.3) and using (3.3), we obtain

Also, substituting $u_1$ and $u_2$ into (1.1), (1.2) and (1.4) and using (3.3) and (3.4), we have the first problem of finding a pair of functions $\{r, u_1\}$ in the form

where

And, for the function $u_2$, we have the second problem in the form

where

4.   Well-posedness of the first problem

4.1.   Existence and uniqueness of the solution of the first problem

Consider the spectral problem

The spectral problem (4.1) has only eigenfunctions

and the eigenvalues are defined by

Since (4.1) is the self-adjoint problem, the system of eigenfunctions $\{y_k(x)\}, \ (k = 1, 2, ...)$ forms an orthonormal basis in $L_2(0, 1)$.

Lemma 4.1. Let $\varphi_1(x)\in C^4[0, 1]$ be a function satisfying the conditions

Then, the following inequality

holds, where $c_1$ is a constant, $\varphi_{1k} = \left(\varphi_1, y_k\right), \ (k = 1, 2, ...)$.

Proof. By using (4.2), integration by parts four times and the Schwarz and Bessel inequalities, we obtain

□

Theorem 4.2. Let the following conditions be satisfied:

$(A_1)$ $\varphi_1\in C^4[0, 1], \ \varphi_1(0) = \varphi_1''(0) = 0, \ \varphi_1(1) = \varphi_1''(1) = 0$;

$(A_2)$ $f_1\in C(\overline{D}_T), \ f_1(\cdot, t)\in C^4[0, 1], \ f_1(0, t) = f_{1xx}(0, t) = 0, \ f_1(1, t) = f_{1xx}(1, t) = 0, \ 0 < \frac{1}{m_0}\leq\min\limits_{0\leq t \leq T}\left|\int_0^1{f_1(x, t)dx}\right|$;

$(A_3)$ $E(t)\in C^1[0, T], \ E(0) = \int\limits_{0}^{1}{\varphi_1(x)}dx$,

where $m_0$ is a constant. Then, the inverse problem (3.5)–(3.8) has a unique classical solution.

Proof. (Existence of the solution of the first problem) To construct a formal solution of problem (3.5)–(3.8), we will use the Fourier method. Following this method, we seek the solution of (3.5)–(3.8) in a Fourier series as

where $u_{1k}(t) = (u_1(\cdot, t), y_k), \ (k = 1, 2, ...)$. For the functions $u_{1k}$ we obtain the Cauchy problem

where $f_{1k}(t) = \left(f_1(\cdot, t), y_k\right), \ \varphi_{1k} = (\varphi_1, y_k), \ (k = 1, 2, ...)$. The solution of this Cauchy problem is given in ^[24] by

where

According to Lemma 2.1, we rewrite (4.3) as

Hence, the formal solution of problem (3.5)–(3.8) is expressed via the series

Now, we get the expression of the term $r$. Integrating Eq (3.5) between 0 and 1, we obtain

By using the overdetermination conditions (3.8) and (3.3), it is easy to deduce that

where

Let us denote

Then, we obtain the Volterra integral equation of the second kind with respect to $r$ in the form

Before we proceed further, notice that, under assumption $(A_2)$, the series

is uniformly convergent, and for $E(t)\in C^{1}[0, T]$ the term $\sum_{i = 1}^{m}q_i \partial^{\alpha_i}_{t}{E}(t)$ is continuous, being the difference of $m$ pieces of continuous functions. According to Lemma 2.2, we estimate the kernel of (4.7) and $L(t)$ in the form

where $m_1$ is a bound of $\sum_{i = 1}^{m}q_i \partial^{\alpha_i}_{t}{E}(t)$ and $c, c_1$ are constants defined in Lemmas 2.2 and 4.1, respectively. Therefore, the kernel of Eq (4.7) is weakly singular and, by Lemma 2.5, there exists a unique solution $r\in C[0, T]$.

Due to assumptions $(A_1)$ and $(A_2)$ in Theorem 4.2, we have

Since the solution $u_1$ is formally given by series (4.5), we need to show that the series corresponding to $u_1, \ u_{1xx}$, and $\sum_{i = 1}^{m}{q_i \partial^{\alpha_i}_{t}u_{1}}$ converge. Under assumptions $(A_1)$–$(A_3)$, for all $(x, t)\in \overline{D}_T$, the series corresponding to $u_1, u_{1xx}$ are bounded from above by the series

where $m_3: = \|r\|_{C\left[0, T\right]}$. Obviously, these majorizing series are convergent.

Now we show that $\sum_{i = 1}^{m}{q_i\partial^{\alpha_i}_{t}u_{1}(x, t)}$ are continuous functions on $D_T$. For this, we first calculate $u'_{1k}(t)$, then we estimate $\partial^{\alpha_1}_{t}u_{1k}(t)$ on $[\varepsilon, T]$ for all $\varepsilon > 0$. The estimates for $\partial^{\alpha_2}_{t}u_{1k}(t), ..., \partial^{\alpha_m}_{t}u_{1k}(t)$ on $[\varepsilon, T]$ are obtained in a similar way.

First, let us get an estimate for $u'_{1k}$. According to Lemma 2.3, we obtain the estimate

Then, for $\partial^{\alpha_1}_{t}u_{1k}$, we have the estimate

where $m_4: = c/(k\pi)^4$. By applying the change of variable $s = \tau/t$ in (4.9), we have

where

Consequently, by (4.10), the functions $\sum_{i = 1}^{m}{q_i\partial^{\alpha_i}_{t}u_{1}(x, t)}$ are continuous on $D_T$. □

Proof. (Uniqueness of the solution of the first problem) Let us show that the solution of problem (3.5)–(3.8) is unique. Suppose that there are two pairs of solutions $\{\widehat{r}, \widehat{u}_1\}$ and $\{\tilde{r}, \tilde{u}_1\}$ of the inverse problem (3.5)–(3.8). Then from (4.5) and (4.7), we have

and

Then, (4.12) yields $\widehat{r} = \tilde{r}$. After substituting $\widehat{r} = \tilde{r}$ in (4.11), we have $\widehat{u}_1 = \tilde{u}_1$. □

4.2.   Continuous dependence of the solution of the first problem on the data

Theorem 4.3. Let $\mathbb{F}$ be the set of triples $\{\varphi_1, f_1, E\}$ where the functions $\varphi_1, f_1$ and $E$ satisfy the assumptions of Theorem 4.2, and

for some positive constants $M_0, \ M_1$ and $M_2$. Then, the solution $(r, u_1)$ of the inverse problem (3.5)–(3.8) depends continuously upon the data on $\mathbb{F}$.

Proof. Let $\mathbb{F} = \{\varphi_1, f_1, E\}$ and $\overline{\mathbb{F}} = \{\overline{\varphi}_1, \overline{f}_1, \overline{E}\}$ be two sets of data, and $\|\mathbb{F}\| = \|f_1\|_{C^{4, 0}\left(\overline{D}_T\right)}+\|\varphi_1\|_{C^4\left[0, 1\right]}+\|E\|_{C^1\left[0, T\right]}$, $(r, u_1)$ and $(\overline{r}, \overline{u}_1)$ be the solutions of the inverse problem (3.5)–(3.8) corresponding to the data $\mathbb{F}$ and $\overline{\mathbb{F}}$, respectively.

Let us denote

For the difference $E-\overline{E}$ we have the estimate

where

First, we write the function $L$, the difference $r-\overline{r}$, and the integral from $0$ to $t$ of the kernel of Eq (4.7) as

Then, from Lemma 2.2 and equality (4.15) we obtain

where $M_4 = m^2_0\left(m_1+c c_1\right), \ M_5 = m_0c\widehat{c}_1, and \ M_6 = m_0M_3$.

By Lemma 2.2, equality (4.14) we arrive at

where $M_7 = m_0 c T^{\alpha_1}/\alpha_1\left(\widehat{c}_2+m_0 c_2\right)$.

According to inequalities (4.17) and (4.18) from (4.16), we have

where

From (4.5), a similar estimate can be obtained for $u_1-\overline{u}_1$. □

5.   Well-posedness of the second problem

To begin with, let us rewrite problem (3.9)–(3.11) as

where

We search for the solution to problem (5.1)–(5.3) in the form

where $a(x) = 20x^7-70x^6+84x^5-35x^4+x$. Consequently, for the unknown function $u_0$ we have the problem

where

The auxiliary spectral problem for the considered direct problem (5.4)–(5.6) is

The spectral problem (5.7) has only eigenfunctions

and the eigenvalues are defined by

Since problem (5.7) is self-adjoint, the system of eigenfunctions $\{\mu_k(x)\}, \ (k = 1, 2, ...)$ forms an orthonormal basis in $L_2(0, 1)$. We consider only the case $\alpha > 0$. The case $\alpha < 0$ will be similar.

For sufficiently large $k$, the asymptotic representation of eigenvalues of problem (5.7) has the form

Lemma 5.1. Let $\varphi_0\in C^4[0, 1]$ be a function satisfying the conditions

Then, the inequality

holds, where $c_3$ is a constant and $\varphi_{0k} = \left(\varphi_0, \mu_k\right), \ (k = 1, 2, ...)$.

Proof. By using (5.8), integration by parts four times, and the Schwarz and Bessel inequalities, we obtain

□

The class of functions which satisfy the conditions of Lemma 5.1 will be denoted by

Similarly, as for the inverse problem (3.5)–(3.8), we search for a solution to problem (5.4)–(5.6) by the Fourier method. Then, we have

where $f_{0k}(t) = \left(f_0(\cdot, t), \mu_k\right), \ \varphi_{0k} = (\varphi_0, \mu_k), \ (k = 1, 2, ...)$.

Theorem 5.2. Suppose $\varphi_0\in \Phi, \ f_0\in C\left(\overline{D}_T\right)$ and $f_0(\cdot, t)\in \Phi$ for every $t\in[0, T]$. Then, (5.9) gives a classical solution $u_0$ to (5.4)–(5.6) and $u_0(\cdot, t)\in C^2[0, 1], \ \partial^{\alpha_i}_{t}{u_0(x, \cdot)}\in C(0, T], \ (i = 1, ..., m)$.

Proof. Let us denote

As in the previous section, we need to show that the series corresponding to $u_0, \ u_{0xx}$, and $\sum_{i = 1}^{m}{q_i \partial^{\alpha_i}_{t}u_{0}}$ converge. The series corresponding to $u_0$ and $u_{0xx}$ are bounded from above by the series

According to Lemma 2.3, we obtain the estimate of $\partial^{\alpha_1}_{t}u_{0k}(t)$ on $[\varepsilon, T]$ for all $\varepsilon > 0$ as

Therefore, $\partial^{\alpha_1}_{t}u_{0k}(t)$ represent the continuous function on $[\varepsilon, T]$ for all $\varepsilon > 0$. □

Remark 5.3. The uniqueness of the solution of problem (5.4)–(5.6), under the conditions of Theorem 5.2 is obtained from the uniqueness of the representation (5.9).

Lemma 5.4. ^[30] Let $u_0$ satisfy Eq (5.4) in $\overline{D}_T$. If $f_0(x, t)\leq 0$ in $\overline{D}_T$, then

If $f_0(x, t)\geq 0$ in $\overline{D}_T$, then

Theorem 5.5. The classical solution of problem (5.4)–(5.6) depends continuously on $\varphi_0\in C[0, 1], \ f_0\in C(\overline{D}_T)$ in the sense that

where $u_0$ and $\overline{u}_0$ are classical solutions of (5.4)–(5.6) with the data $f_0, \ \varphi_0$, and $\overline{f}_0, \ \overline{\varphi}_0$, respectively.

Proof. Let $u_0$ be a classical solution of problem (5.4)–(5.6). We introduce the following:

Let us construct the function

The function $\omega$ is the classical solution of problem

Using the maximum principle, we obtain the estimates

and

Similarly, if we introduce the function

and using the minimum principle we arrive at the opposite estimate

Hence, if $u_0$ is the classical solution of problem (5.4)–(5.6), we have the estimate

To prove the continuous dependence on the data, we study the difference $g(x, t) = u_0(x, t) - \overline{u}_0(x, t)$. This function is the classical solution of (5.4)–(5.6) with $f_0 - \overline{f}_0$ and $\varphi_0 - \overline{\varphi}_0$ replaced by $f_0$ and $\varphi_0$, respectively. Applying inequality (5.11) to $g$, we arrive at estimate (5.10). □

As a consequence of Theorem 5.2, we obtain the following main theorem for problem (5.1)–(5.3):

Theorem 5.6. Let $\varphi_2\in C^4[0, 1], \ f_2\in C\left(\overline{D}_T\right), \ f_2(\cdot, t)\in C^4[0, 1]$ for every $t\in[0, T]$, and the following conditions

be satisfied. Then, the classical solution $u_2$ of problem (5.1)–(5.3) exists, is unique and $u_2(\cdot, t)\in C^2[0, 1], \ \partial^{\alpha_i}_{t}{u_2(x, \cdot)}\in C(0, T], \ (i = 1, ..., m)$.

Proof. The solution to problem (5.1)–(5.3) has the form

It can be seen from (5.13) that the majorizing series for (5.13) and (5.9) are the same and these series converge when conditions (5.12) are met. The uniqueness comes from the fact that the homogeneous problem (5.1)–(5.3) (that is, when $\varphi_2, \ \gamma, \ rf_2\equiv0$) has only a trivial solution. □

6.   The main results

Theorem 6.1. Let the following conditions be satisfied:

$(H_1)$ $\varphi\in C^1[0, 1], \ \varphi(0)+\varphi(1) = 0, \ a_1\varphi'(0)+b_1\varphi'(1)+a_0\varphi(0) = 0$;

$(H_2)$ $f\in C\left(\overline{D}_T\right), \ f(\cdot, t)\in C^1[0, 1], \ f(0, t)+f(1, t) = 0, \ f_x(0, t)+f_x(1, t)+\alpha\left(f(0, t)-f(1, t)\right) = 0$;

$(H_3)$ $E\in C^1[0, T], \ E(0) = \int\limits_{0}^{1}{\varphi(x)}dx$.

Then the classical solution $u$ of problem (1.1)–(1.4) exists, is unique and $u(\cdot, t)\in C^2[0, 1], \ \partial^{\alpha_i}_{t}{u(x, \cdot)}\in C(0, T], \ (i = 1, ..., m)$.

Proof. The smoothness conditions specified in assumptions $(H_1)$–$(H_3)$ are derived from Theorems 4.2 and 5.6. To prove the consistency conditions in Theorem 6.1, it is enough for us to prove the following lemma. □

Lemma 6.2. Let assumptions $(H_1)$–$(H_3)$ be fulfilled. Then, the conditions

are satisfied.

Proof. It is easy to see from the representation $\gamma$ that

Substituting the expression $\varphi(x) = \varphi_1(x)+\varphi_2(x)$ into $\varphi(0)+\varphi(1) = 0$, we have

Using the conditions $\varphi_2(0)+\varphi_2(1) = 0$, $\varphi_1(0) = \varphi_1(1)$ from (6.2), we obtain $\varphi_1(0) = \varphi_1(1) = 0$. Also, substituting the expression $\varphi(x) = \varphi_1(x)+\varphi_2(x)$ into $a_1\varphi'(0)+b_1\varphi'(1)+a_0\varphi(0) = 0$, we have

The conditions $\varphi_1'(0) = -\varphi_1(1), \ \varphi_1(0) = 0, \ \varphi_2'(0) = \varphi'_2(1)$ imply that

which, from (6.1), becomes $\varphi_2'(0)+\alpha\varphi_2(0) = \gamma(0)$.

Similarly, we get $\varphi_2'(1)-\alpha\varphi_2(1) = \gamma(0)$.

Substituting the expression $f(x, t) = f_1(x, t)+f_2(x, t)$ into $f(0, t)+f(1, t) = 0$, we have

Then using conditions $f_1(0, t) = f_1(1, t), \ f_2(0, t)+f_2(1, t) = 0$ from (6.3), we arrive at $f_1(0, t) = f_1(1, t) = 0$.

Applying $f(x, t) = f_1(x, t)+f_2(x, t)$ and the conditions $f_1(0, t) = f_1(1, t), \ f_2(0, t)+f_2(1, t) = 0, \ f_{1x}(0, t)+f_{1x}(1, t) = 0, \ f_{2x}(0, t) = f_{2x}(1, t)$ from $f_x(0, t)+f_x(1, t)+\alpha\left(f(0, t)-f(1, t)\right) = 0$, we obtain $f_{2x}(0, t)+\alpha f_{2}(0, t) = \frac{b(t)}{r(t)}$.

Similarly, we have $f_{2x}(1, t)-\alpha f_{2}(1, t) = \frac{b(t)}{r(t)}$.

Substituting the expression $\varphi(x) = \varphi_1(x)+\varphi_2(x)$ into $(H_3)$, we obtain $E(0) = \int\limits_{0}^{1}{\varphi_2(x)}dx$.

The uniqueness of the solution of problem (1.1)–(1.4) comes from the fact that $u_1$ and $u_2$ are unique. This completes the proof. □

Remark 6.3. If we consider the direct problem (1.1)–(1.3), it is easy to see that, for the direct problem (1.1)–(1.3), we have a theorem of existence and uniqueness, which is similar to Theorem 6.1 without assumption $(H_3)$.

7.   Conclusions

In this paper, we considered the inverse source problem with the nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. Since the eigenvalues of the auxiliary spectral problem do not form a basis, we have divided the problem into two sub-problems, one of which is the inverse problem, and the second the direct problem. The well-posedness of the inverse and direct problems are shown by Fourier expansion in terms of eigenfunctions of the corresponding spectral problems. Also, for the well-posedness of the inverse problem, the properties of the Volterra integral equation of the second kind were used. The continuous dependence on the data of the solutions of the inverse and direct problems was proved.

Since a feature of this paper is the nonlocal boundary condition (1.3), performing some numerical tests is more difficult even without the fractional derivative (see for example ^[31]). This is due to the fact that the sufficient conditions for the existence of a solution are not satisfied using standard methods. It is also difficult to show the convergence of the series that arises when solving (1.1)–(1.4) using the Fourier method.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.

Acknowledgements

The authors are grateful to the anonymous reviewers for their careful reading of the article, as well as for many useful comments and suggestions that helped improve the presentation of the article.