An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data. A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization parameters. +e numerical examples illustrate the behavior of the proposed method.

However, in some practical situations, part of boundary data, or initial data, or diffusion coefficient, or source term, or the order of fractional derivative may not be given and we want to find them by additional measured data which will yield some inverse problems of the fractional diffusion equation.
e early papers on inverse problems of timefractional diffusion equation were provided by Murio in [25,26] for solving the time-fractional diffusion equation by mollification methods. Cheng et al. [27] gave the uniqueness in determining the parameter α and p(x) by means of observation data at one end point. Liu and Yamamoto [28] considered a backward problem in time for a time-fractional partial diffusion equation in one-dimensional case. Zhang and Xu [29] studied an inverse source problem in the timefractional diffusion equation and proved uniqueness for identifying a space-dependent source term by using analytic continuation and Laplace transform. Zheng and Wei [30] applied a new regularization method to solve an inverse problem for a time-fractional diffusion equation in a onedimensional semi-infinite domain. Jin and Rundell [31] studied an inverse problem of recovering a spatially varying potential term in the 1D time-fractional diffusion equation from the flux measurements and proposed a quasi-Newtontype reconstruction algorithm. Recently, Wei and Wang [32] proposed a modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Wei et al. [33] identified a time-dependent source term in a multidimensional time-fractional diffusion equation from boundary Cauchy data.
As for time-fractional inverse diffusion problem for onedimensional (1D) models, theoretical concepts and computational implementation have been discussed by many authors, and a number of solution methods have been proposed, i.e., spectral regularization method [34], iteration regularization method [35], optimal regularization method [36], and a new dynamic method [37]. However, for the twodimensional time-fractional inverse diffusion problem, few results are available. Xiong et al. [38] gave a conditional stability estimate for the inverse heat conduction problem in the 2D time-fractional heat equation and studied a dynamic spectral regularization method with numerical testification. However, error estimates were given in [38] by only choosing the regularization parameter by a priori choice rule.
In the present paper, we will consider the 2D time-fractional inverse diffusion problem by using a modified kernel method and a posteriori parameter choice rule is given. For a priori choice and a posteriori choice of the regularization parameter, we obtain the convergence estimates, respectively. e rest of this paper is organized as follows. In Section 2, we describe the 2D time-fractional inverse diffusion problem and give an analysis on the ill-posedness of this problem. In Section 3, we propose a modified kernel method and prove the convergence estimates under a priori and a posteriori parameter choice rule, respectively. Section 4 is the numerical aspect of the proposed method. Finally, we give a brief conclusion in Section 5.

Description of the 2D Time-Fractional Inverse Diffusion Problem
We consider the following 2D time-fractional inverse diffusion problem: with the corresponding measured data function g δ (y, t) and initial and boundary conditions where the time-fractional derivative (z α u/zt α ) is the Caputo fractional derivative of order α(0 < α ≤ 1) defined by (see [39]) Here, we wish to determine the temperature u(x, y, t) for 0 ≤ x < 1 from the temperature measurements g δ (y, t).
In order to apply the Fourier transform techniques, we extend the functions u(x, ·, ·), g(·, ·), and g δ (·, ·) to the whole plane − ∞ < y < +∞ and − ∞ < t < +∞ by setting the functions to be zero for y < 0 or t < 0. We assume that these functions are in L 2 (R 2 ) and use the corresponding L 2 -norm, defined as follows: We also assume that the measured data function g δ (y, t) satisfies where the constant δ > 0 represents a bound on the measurement error. Let be the Fourier transform of a function g(y, t). Taking the transform to equations (1)-(6) with respect to y and t, we can get the solution of equations (1)- (6) in the frequency domain (see [38]): where Denote c as follows: 2

Mathematical Problems in Engineering
For the above problem, since |e (1− x) ����� (iη) α +ξ 2 √ | is unbounded with respect to variables ξ and η for fixed 0 ≤ x < 1, the small error in the high-frequency components will be amplified. erefore, the 2D time-fractional inverse diffusion problem is severely ill-posed. To solve the 2D time-fractional inverse diffusion problem, a natural way to stabilize the problem is eliminate the high frequencies or to replace the "kernel" e (1− x) by a bounded approximation.

A Modified Method and Convergence Estimates
In this section, we will give a modified "kernel" method and obtain the convergence estimates. e regularization solution is given by where where β(x) > 1 is the regularization parameter.
To obtain the convergence estimates between the regularization solution and the exact solution, we need to assume a priori bound: where E > 0 is a constant and ‖·‖ p denotes the norm in Sobolov space H p (R 2 ) defined by Remark 1. Obviously, when p � 0, we can know that (17) is bounded in the L 2 (R 2 )-norm.

A Priori Parameter Choice.
In the following, we give the convergence estimate for ‖u δ β (x, ·, ·) − u(x, ·, ·)‖ by using a priori choice rule for the regularization parameter.
is the regularization solution with noisy data g δ (y, t) and that u(x, y, t) is the exact solution with the exact data g(y, t). Let assumption (9) be satisfied and let ‖u(0, ·, ·)‖ ≤ E. If we choose then for every x ∈ (0, 1), we obtain the error estimate Proof. Due to Parseval's identity and triangle inequality, we can obtain □ For I 1 , we have From (11), we know We now estimate I 2 ; note that from (23), Let Differentiating φ(A) and setting the derivative equal to zero, we find that is the maximum value point of the function φ(A). erefore, we know Combining (19), (21), (22), and (27), we have the Hölder-type error estimate (20).

Mathematical Problems in Engineering
e error estimate in eorem 1 does not give any useful information on the continuous dependence of the solution at x � 0. To retain the continuous dependence of the solution at x � 0, one has to introduce a stronger a priori assumption.

A Posteriori Parameter Choice.
In this section, we first give the following lemma.
Lemma 1 (see [40]). Let the function f(λ): (0, a] ⟶ R be given by with a constant c ∈ R and positive constants a < 1, b, and d; then, for the inverse function, we have In this following, we give the convergence estimate for ‖u δ β (x, ·, ·) − u(x, ·, ·)‖ by using a posteriori choice rule for the regularization parameter, i.e., Morozov's discrepancy principle.
According to Morozov's discrepancy principle [41], we adopt the regularization parameter β as the solution of the following equation: where τ > 1 is a constant. Denote Here and in the following, β is a function of the variable x and sometimes the function β(x) is also used.

(45)
Proof. Due to the triangle inequality and (43), note that |T β | ≤ 1, and we obtain We now estimate the second term on the right hand side of (46): Similar to the proof of eorem 2, we can know Combining (46) and (48), we obtain

Lemma 4. If β(x) is the solution of equation
Proof. Using the triangle inequality and (43), we have □ Theorem 3. Assume that conditions (9) and (17) Proof. Due to the Parseval formula and the triangle inequality, we obtain From Lemma 3, we can know that (1)).

Numerical Aspect
In this section, we present two numerical examples intended to illustrate the behavior of the proposed method.
e numerical examples are constructed in the following way. First, we present the initial data u(0, y, t) � f(y, t) of 2D time-fractional diffusion problem at x � 0 and computed the function u(1, y, t) � g(y, t) by solving a direct problem, which is a well-posed problem. en, we added a random distributed perturbation to the data function obtaining vector g δ (y, t), i.e., g δ (y, t) � g(y, t) + ε randn(size(g(y, t))), where g(y, t) � g y 1 , t 1 , g y 2 , t 2 , . . . , g y n , t n T , where δ indicates the error level of g, i.e., In numerical implementations, we give the data f(y, t) and sample at an equidistant grid in the domain [0, 1] × [0, 1] with 64 × 64 grid points. e function "randn(·)" generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ 2 � 1, and standard deviation σ � 1; "randn size(g(y, t))" returns an array of random entries which has the same size as g(y, t). Let RMS denotes the mean square for a sampled function φ(·, ·) which is defined by where n is the total number of test points. Similarly, we can define the mean square error (RMSE) between the computed data and the exact data. Finally, we solved the 2D time-fractional inverse diffusion problem by the modified method. (66)  Table 1 shows the comparison of RMSE of Example 1 for different x with ε � 0.0001 and α � 0.1. We find that the smaller the x is, the worse the computed approximation will be. Table 2 shows the comparison of RMSE of Example 1 for different ε with α � 0.1 at x � 0.5. e numerical error is decreasing as the level of noise becomes smaller. Table 3 shows the comparison of RMSE of Example 1 for different α with ε � 0.0001 at x � 0.5. e numerical accuracy is stable to the fractional order α.

Conclusion
In this paper, we propose a modified method to solve the time-fractional inverse diffusion problem in the two-dimensional setting. For the choice of regularization parameter, we give not only a priori but also a posteriori rules. Moreover, under both a priori rule and a posteriori rule, we prove the error estimates from the viewpoint of theoretical analysis.
e numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In general, E > 0 such that a priori condition (17) holds will not be known. But such a E has to be known if one wants to construct a priori parameter choice rule. We propose a posteriori parameter choice rule which is independent of E. So it is more implementable in practical application. Moreover, the numerical experiments show that a posteriori parameter choice rule also works well.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.