# Determination of an Energy Source Term for Fractional Diffusion Equation

Authors: Sid Ahmed Ould Ahmed Mahmoud; Hamed Ould Sidi; Maawiya Ould Sidi
Journal: Journal of Sensors
 (2022)

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## Abstract

In this article, we will determine the source term of the fractional diffusion equation (FDE). Our contribution to this work is the generalization of the common inverse diffusion equation issues and the inverse diffusion equation problems for fractional diffusion equations with energy source and using Caputo fractional derivatives in time and space. The problem is reformulated in a least-squares framework, which leads to a nonconvex minimization problem, which is solved using a Tikhonov regularization. By considering the direct problem with an implicit finite difference scheme (IFDS), the numerical inversions are performed for the source term in several approximate spaces. The inversion algorithm (IA) uniqueness is obtained. Furthermore, the effect of fractional order and regularization parameter on the inversion algorithm is carried out and shows that the inversion algorithm is effective. The order of fractional derivatives expresses the global property of the direct problem and also shows the badly posed nature of the inverted problem in question.


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## Body

## 1. Introduction

Describing and simulating the behavior and diffusion process of contaminants in a heterogeneous porous medium are essential for environmental protection. The advection-dispersion equation with whole-order derivatives has played an important role in modeling the diffusion of contaminants over the past forty years, known as the classical diffusion model. However, some research has shown that the classical model is insufficient to present many real problems, where a particle plume propagates faster or slower than expected by the whole-order scattering equation. In the saturated zone of a very heterogeneous aquifer, Adams and Gelhar in [1] showed that the field data are not well simulated by the classical advection-diffusion equation, and the results indicate slower diffusion than normal. Slow diffusion is formulated by the long-tailed spatial density distribution position as time passes, which has been considered and studied in many domains of applied science. Hatano and Hatano in [2] consider dispersive transport of ions in column experiments. Giona et al. in [3] study the relaxation in a complex viscoelastic material. Berkowitz et al. in [4] discuss the anomalous transport heterogeneous porous media. Zhou and Selim in [5] propose an application of the fractional advection-dispersion equation in porous media. Xiong et al. in [6] modeled the transport of the solute in a homogeneous one-dimensional model. Jajarmi et al. in [7] propose a fractional formulation for immunogenic tumor. Baleanu et al. in [8] propose a new comparative study on the general fractional model. Erturk et al. in [9] give a new fractional-order Lagrangian to describe a Beam motion. The slow diffusion called abnormal underdiffusion can be modeled in mathematics by the following fractional temporal diffusion equation:
(1)∂γu∂γtx,t=∂∂x∂u∂xx,t+rx,t,0<x<l,0<t<T,0<γ<1.There is some theoretical and numerical research on the direct problem for the FDE with initial and boundary conditions. For the theoretical study, Gorenflo et al. in [10] treat the solvability of linear fractional differential equations. Metzler et al. in [11] consider value problems for fractional diffusion equations. Numerical approaches are proposed by Meerschaert and Tadjeran in [12] and Liu et al. in [13]. However, inverse problems governed by the fractional equation get less attention in the literature. According to Podlubnv [14], let us consider an inverse recovery boundary function problem from a data set at an internal point of a semi-infinite one-dimensional temporal fractional equation. A stable algorithm established and analyzed by modification methods. Cheng et al. [15] have investigated the inverse problem of finding the fractional order and the space-dependent diffusion coefficient in equation (1). By additional boundary data, they prove the uniqueness of the inverse problem in the case of null Neumann boundary conditions. Recently, much research has focused on the theory and applications of inverse problems. Li et al. in [16] treat numerical inversions in a fractional equation. Zhang and Xu in [17] consider an inverse problem for a fractional equation. Li et al. in [18] study the stability of the inverse acoustic problem. Jiang et al. in [19] use an inverse source problem with a hyperbolic equation. Maisto et al. in [20] propose resolution limits for strip currents. Can et al. in [21] treat the inverse problem with Mittag-Leffler. Liu et al. in [22] propose a method for solving a nonlinear wave inverse energy problem. Smirnov et al. in [23] consider an inverse problem for the radiative transfer equation. Slodicka et al. in [24] study the uniqueness for an inverse source problem of determining a space dependent source in a time-fractional equation. Le et al. in [25] solve numerically an inverse problem for hyperbolic equations. Bao et al. in [26] study stability for inverse problems in elastic and electromagnetic waves. Sun and Liu in [27] consider an inverse problem for distributed order time-fractional equations. Yang et al. in [28] propose an iterative method for solving the inverse problem of the time-fractional wave equation. Qiu et al. in [29] give a novel homogenization function method for an inverse problem. Cheng and Liu in [30] study a source problem with local measurements. Hrizi et al. in [31] give a new method for a parabolic inverse problem. Tuan et al. in [32] study an identification of inverse source for fractional equation.Our primary contribution to this work is the generalization of the common inverse diffusion equation issues and the inverse diffusion equation problems for fractional diffusion equations with energy source and using the Caputo fractional derivatives in time and space. In recent years, a great deal of work has been done to examine these mathematical problems, both theoretically and numerically. These mathematical problems have become a significant tool in simulating many real-life difficulties. For recent developments, one can see Hendy, and Bockstal in [33] for reconstruction of a solely time-dependent source in a time-fractional equation. Ansari et al. in [34] study a class of distributed order fractional diffusion equation. Jajarmi et al. in [7] propose a general fractional formulation and tracking control for a tumor. Erturk et al. in [9] use novel fractional-order Lagrangian to describe the motion of beam.In this paper, we investigated the inverse problem to numerically determine the energy source term for Caputo temporal FDE. In a one-dimensional space, we apply an algorithm based on the least-squares method with Tikhonov regularization. An IFDS is used for the direct problem. Numerical inversions of energy source terms dependent on space are performed in different approximation spaces using numeric inversion. We show the numerical uniqueness of the IA. The inversion results show the effectiveness of the IA for this inverse problem.In the first part, we consider nonlinear square problem, which are ill-posed and have many solutions and do not give good simulation results. In the second part, we use the Tikhonov regularization, which allows for successful simulations. In Section2, we provide preliminary results, then study the last square problem, and discuss the existence and uniqueness of the solution to the Tikhonov regularization problem. We examine the solution to the final square problem. Section 3 shows an inversion algorithm that will be tested successfully through three examples of simulations in Section 4.


## 2. Well-Posedness

Let consider the following system(2)∂γu∂γtx,t−∂∂xkx∂u∂xx,t=rx,t,0<x<l,0<t<T,0<γ<1.(3)ux,0=u0,0≤x≤l,(4)u0,t=φt,uxl,t=gt,0≤t≤T,wherel,T>0,u0,k∈C10,l,φ,g∈C0,T,k≥0 in 0,l and ∂γu/∂γt, is the fractional Caputo derivative defined by
(5)∂γu∂γt=CDtγux,t=1Γ1−γ∫0T∂ux,s∂sdst−sγ,where (3) and the initial condition (4) are the Dirichlet-Neumann boundary conditions, rx,t is the energy perturbation term, and kx is the spatial diffusion coefficient. The fractional diffusion problem (2)-(4) is a mathematical model to describe the phenomenon of contaminant flux in a medium heterogeneous porous, see Zhou and Selim in [5] and Xiong et al. in [6].Theorem 1.
Let1/2<γ<1, and Ω=0,l. If r∈L20,T;Ω, then, for any u0∈L2Ω, the problem (2) has a unique solution u∈L20,T;Ω. Moreover
(6)uL20,T;Ω≤KΓγ2T2γ−12γ−1u0X+2T3γγ3rL20,T;Ω.
Now, we considerrx,t=Dxft.In the case where the sourceDxis not defined, we look for it by additional observations at the right end ofx=l, we search for it by additional observationsx=l,i.e., it is a limit.
(7)ul,t=ψt,0<t≤T.
So, the energy problem is defined by the FDE (3), with the condition defined by (3) - (5) and an additional condition (7).
To find the solution of the energy problem, the functionD must be parameterized in the following polynomial form. For x∈Ω, we define the family of functions pmx:m=1,2,⋯, with the following approximate expansion:
(8)px=∑i=0mpixi.
For anyDx, a unique solution of a corresponding direct problem, indicated by ux,t;D, can be developed using the finite difference scheme in [16], then ul,t;D is obtained. Therefore, getting a term source D amounts to finding a vector P=p0,p1,⋯,pmT, which means that one can write ul,t;D=ul,t;P.


### 2.1. Nonlinear Least Squares Problem

For finding the solution of the energy problem, we solve the following nonlinear least squares problem(9)minPΦP,where
(10)ΦP=ul,t;P−ψt22,0<t≤T.The problem (9) is badly posed so that this problem admits several solutions. For uniqueness, we used Tikhonov’s regularization.


### 2.2. Tikhonov Regularization

We consider the following regularized problem(11)minPΦαP,where
(12)ΦαP=ul,t;P−ψt22+αP22,such thatα≥0 is the regularization parameter. Now to get Dj=Dxj,j=0,1,2,⋯, we assume that
(13)Dj+1=Dj+δDj,j=0,1,2,⋯,where j is the number of iterations. δDj denotes a perturbation of Dj.From (8), we have
(14)Djx=∑i=0mpijxi,(15)δDjx=∑i=0mδpijxi,with δPj denotes a perturbation for a given Pj. So, to get Pj+1, from given Pj, it suffices to obtain a disturbance δPj.(16)Pj+1=Pj+δPj,j=0,1,2,⋯.Therefore, we just have to find a disturbance vector.(17)δPj=δp0j,δp1j,⋯,δpmjT.In the following, to make writing easier,Pj and δPj are abbreviated as P and δP, respectively.


### 2.3. Linear Problem

We have the direct solutionul,t;P+δP is implicitly dependent on P, so with the Taylor expansion to order one, we find
(18)ul,t;P+δP≈ul,t;P+∇pTul,t;P.δp.Using (12), we get
(19)ΦP+δP=ul,t;P+δP−ψt22.Hence, in view to (18), we get
(20)ΦP+δP=∇pTul,t;P.δp−ul,t;P+δP−ψt22.From (12), the objective least squares function becomes
(21)FαδP=∇PTul,t;P.δp−ul,t;P+δP−ψt22+αP22.We approximate the domain0,T, by tnn=1,2,⋯,N, and 0=t1<t2<⋯<tN=T. Using the finite difference method, we have
(22)∇pTul,tn;P.δ≈∑i=0mul,tn;p0,⋯,pi+τ,⋯,pm−ul,tn;PτδPi,forn=1,2,⋯,N, where τ is a digital differential step. Hence, we define the matrix
(23)H=hniN×m+1,where
(24)hni=ul,tn;p0,⋯,pi+τ,⋯,pm−ul,tn;P.Thus(25)H=h10h11⋯h1mh20h21⋯h2m⋮⋮⋱⋮hN0hN1⋯hNm.Let(26)U=ul,t1;P,ul,t2;P,⋯,ul,tN;PT,Ψ=ψt1,ψt2,⋯,ψtNT.Using (22), (25), and (26), we can write (21) in the following form.
(27)FαδP=Hδp−Ψ−U22+αP22.Lemma 2.
TheδPα is a minimum of Fα, if and only if δPα solves the following normal equation
(28)αδPα+HTHδPα=HTΨ−U.Proof.
(29)FαδP−FαδPα=HδP−Ψ−U22+αδP22−HδPα−Ψ−U22+αδPα22−HδP−Ψ−U22−2<HδP,Ψ−U>+αδP22=HδP−δPα22−2HδPα22+2<HδP,HδPα>−2<HδP,Ψ−U>+2<HδPα,Ψ−U>+αδP−δPα22−2αδPα22+2α<δP,δPα>=2<HδPα−Ψ−U,HδP−δPα>+2<δPα,δP−δPα>=HδP−δPα22+αδP−δPα22=2<HTHδPα−Ψ−U+αδPα,δP−δPα>+HδP−δPα22+αδP−δPα22.
For allδP,δPα∈ℝm+1. If δPα satisfies (27), then, FαδP−FαδPα≥0, thus, δPα minimizes (28). On the other hand, if δPα minimizes (27), choose δP=δPα+tZ, for all t>0, and Z∈ℝm+1, we have
(30)FαδP−FαδPα≥0.
Hence(31)2t<HTHδPα−Ψ−U+αδPα,Z>+t2HZ22+α2Z22≥0.
Divide byt>0, and when t⟶0, we get
(32)<HTHδPα−Ψ−U+αδPα,Z>≥0,forallZ∈ℝm+1.
If we chooseZ=−HTHδPα−Ψ−U+αδPα, we get
(33)<HTHδPα−Ψ−U+αδPα,−HTHδPα−Ψ−U+αδPα>≥0,this implies that
(34)HTHδPα−Ψ−U+αδPα=0,which means that δPα solves the normal equation (28).


### 2.4. Existence Result

Theorem 3.
There exists at leastδPα∈ℝm+1, minimum point of the function Fα.Proof.
LetδPn∈ℝm+1 be a minimizing sequence, i.e.,
(35)FαδPn⟶I=infδP∈ℝm+1FαδP.when n tends to infinity. We show that δPn is a Cauchy sequence
(36)FαδPn+FαδPm=HδPn−Ψ−U22+αδPn22+HδPm−Ψ−U22+αδPm22=HδPn22+Ψ−U22−2<HδPn,Ψ−U>+αδPn22+HδPm22+Ψ−U22−2<HδPm,Ψ−U>+αδPm22=HδPn22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+αδPn22+HδPm22+αδPm22.
However, we have(37)HδPn22+HδPm22=12HδPn+δPm22+HδPn−δPm22,(38)αδPm22+δPm22=α2δPn+δPm22+δPn−δPm22.
Therefore, we may write(39)FαδPn+FαδPm=12HδPn+δPm22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+α2δPn+δPm22+12HδPn+δPm22+α2δPn−δPm22.
On the other hand, we have(40)2Fα12δPn+δPm=H12δPn+δPm−Ψ−U22+2α12δPn+δPm22=12HδPn+δPm−2Ψ−U22+α2δPn+δPm22=12HδPn+δPm22+4Ψ−U22−4<HδPn+δPm,Ψ−U>+α2δPn+δPm22=12HδPn+δPm22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+α2δPn+δPm22.s
By replacing (39) in (40), we get
(41)FαδPn+FαδPm=2Fα12δPn+δPm+12HδPn−δPm22+α2δPn−δPm22≥2I+α2δPn−δPm22.
By taking the limit asn and m to infinity, we get FαδPn+FαδPm⟶2I, thus
(42)δPn−δPm22⟶0asn,m⟶∞.
This shows thatδPn is a Cauchy sequence and therefore convergent since the Euclidean space is a Hilbert space. Let δPα=limn⟶∞δPn, we have δPα∈ℝm+1, by taking into account that Fα is continuous, we conclude that FαδPn⟶FαδPα, then, δPα=I. That proves the existence of a minimum of Fα.


### 2.5. Uniqueness Result

Lemma 4.
Fα admits a unique minimum δPα∈ℝm+1. This minimum δPα is the unique solution of the normal equation
(43)αδPα+HTHδPα=HTψ−U.Proof.
The question is to prove the uniqueness of the solution for the normal equation, because we have an equivalence between (27) and (28). We observe that the normal will be written as
(44)αI+HTHδPα=HTψ−U.
In order to show that this equation admits a unique solution, we must prove that the matrixαI+HTH is positive definite. Indeed, ∀δP≠0,(45)<αI+HTHδP,δP>=<αδP,δP>+<HTHδP,δP>+αδP22+HδP22.
Hence(46)<αI+HTHδP,δP>>0.
This proves that the matrix is positive definite, and so invertible, we deduce that equation (28) admits a unique solution, and therefore, Fα admits a unique minimum. Thus, a perturbation can be determined by (28)
(47)δPα=αI+HTH−1HTΨ−U.
Therefore, the solution can be characterized by (12) with a sufficiently large iteration numbers, or when the perturbation verified the convergence condition
(48)δPα2≤ε,for a given convergence precision ε.


## 2.1. Nonlinear Least Squares Problem

For finding the solution of the energy problem, we solve the following nonlinear least squares problem(9)minPΦP,where
(10)ΦP=ul,t;P−ψt22,0<t≤T.The problem (9) is badly posed so that this problem admits several solutions. For uniqueness, we used Tikhonov’s regularization.


## 2.2. Tikhonov Regularization

We consider the following regularized problem(11)minPΦαP,where
(12)ΦαP=ul,t;P−ψt22+αP22,such thatα≥0 is the regularization parameter. Now to get Dj=Dxj,j=0,1,2,⋯, we assume that
(13)Dj+1=Dj+δDj,j=0,1,2,⋯,where j is the number of iterations. δDj denotes a perturbation of Dj.From (8), we have
(14)Djx=∑i=0mpijxi,(15)δDjx=∑i=0mδpijxi,with δPj denotes a perturbation for a given Pj. So, to get Pj+1, from given Pj, it suffices to obtain a disturbance δPj.(16)Pj+1=Pj+δPj,j=0,1,2,⋯.Therefore, we just have to find a disturbance vector.(17)δPj=δp0j,δp1j,⋯,δpmjT.In the following, to make writing easier,Pj and δPj are abbreviated as P and δP, respectively.


## 2.3. Linear Problem

We have the direct solutionul,t;P+δP is implicitly dependent on P, so with the Taylor expansion to order one, we find
(18)ul,t;P+δP≈ul,t;P+∇pTul,t;P.δp.Using (12), we get
(19)ΦP+δP=ul,t;P+δP−ψt22.Hence, in view to (18), we get
(20)ΦP+δP=∇pTul,t;P.δp−ul,t;P+δP−ψt22.From (12), the objective least squares function becomes
(21)FαδP=∇PTul,t;P.δp−ul,t;P+δP−ψt22+αP22.We approximate the domain0,T, by tnn=1,2,⋯,N, and 0=t1<t2<⋯<tN=T. Using the finite difference method, we have
(22)∇pTul,tn;P.δ≈∑i=0mul,tn;p0,⋯,pi+τ,⋯,pm−ul,tn;PτδPi,forn=1,2,⋯,N, where τ is a digital differential step. Hence, we define the matrix
(23)H=hniN×m+1,where
(24)hni=ul,tn;p0,⋯,pi+τ,⋯,pm−ul,tn;P.Thus(25)H=h10h11⋯h1mh20h21⋯h2m⋮⋮⋱⋮hN0hN1⋯hNm.Let(26)U=ul,t1;P,ul,t2;P,⋯,ul,tN;PT,Ψ=ψt1,ψt2,⋯,ψtNT.Using (22), (25), and (26), we can write (21) in the following form.
(27)FαδP=Hδp−Ψ−U22+αP22.Lemma 2.
TheδPα is a minimum of Fα, if and only if δPα solves the following normal equation
(28)αδPα+HTHδPα=HTΨ−U.Proof.
(29)FαδP−FαδPα=HδP−Ψ−U22+αδP22−HδPα−Ψ−U22+αδPα22−HδP−Ψ−U22−2<HδP,Ψ−U>+αδP22=HδP−δPα22−2HδPα22+2<HδP,HδPα>−2<HδP,Ψ−U>+2<HδPα,Ψ−U>+αδP−δPα22−2αδPα22+2α<δP,δPα>=2<HδPα−Ψ−U,HδP−δPα>+2<δPα,δP−δPα>=HδP−δPα22+αδP−δPα22=2<HTHδPα−Ψ−U+αδPα,δP−δPα>+HδP−δPα22+αδP−δPα22.
For allδP,δPα∈ℝm+1. If δPα satisfies (27), then, FαδP−FαδPα≥0, thus, δPα minimizes (28). On the other hand, if δPα minimizes (27), choose δP=δPα+tZ, for all t>0, and Z∈ℝm+1, we have
(30)FαδP−FαδPα≥0.
Hence(31)2t<HTHδPα−Ψ−U+αδPα,Z>+t2HZ22+α2Z22≥0.
Divide byt>0, and when t⟶0, we get
(32)<HTHδPα−Ψ−U+αδPα,Z>≥0,forallZ∈ℝm+1.
If we chooseZ=−HTHδPα−Ψ−U+αδPα, we get
(33)<HTHδPα−Ψ−U+αδPα,−HTHδPα−Ψ−U+αδPα>≥0,this implies that
(34)HTHδPα−Ψ−U+αδPα=0,which means that δPα solves the normal equation (28).


## 2.4. Existence Result

Theorem 3.
There exists at leastδPα∈ℝm+1, minimum point of the function Fα.Proof.
LetδPn∈ℝm+1 be a minimizing sequence, i.e.,
(35)FαδPn⟶I=infδP∈ℝm+1FαδP.when n tends to infinity. We show that δPn is a Cauchy sequence
(36)FαδPn+FαδPm=HδPn−Ψ−U22+αδPn22+HδPm−Ψ−U22+αδPm22=HδPn22+Ψ−U22−2<HδPn,Ψ−U>+αδPn22+HδPm22+Ψ−U22−2<HδPm,Ψ−U>+αδPm22=HδPn22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+αδPn22+HδPm22+αδPm22.
However, we have(37)HδPn22+HδPm22=12HδPn+δPm22+HδPn−δPm22,(38)αδPm22+δPm22=α2δPn+δPm22+δPn−δPm22.
Therefore, we may write(39)FαδPn+FαδPm=12HδPn+δPm22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+α2δPn+δPm22+12HδPn+δPm22+α2δPn−δPm22.
On the other hand, we have(40)2Fα12δPn+δPm=H12δPn+δPm−Ψ−U22+2α12δPn+δPm22=12HδPn+δPm−2Ψ−U22+α2δPn+δPm22=12HδPn+δPm22+4Ψ−U22−4<HδPn+δPm,Ψ−U>+α2δPn+δPm22=12HδPn+δPm22+2Ψ−U22−2<HδPn+δPm,Ψ−U>+α2δPn+δPm22.s
By replacing (39) in (40), we get
(41)FαδPn+FαδPm=2Fα12δPn+δPm+12HδPn−δPm22+α2δPn−δPm22≥2I+α2δPn−δPm22.
By taking the limit asn and m to infinity, we get FαδPn+FαδPm⟶2I, thus
(42)δPn−δPm22⟶0asn,m⟶∞.
This shows thatδPn is a Cauchy sequence and therefore convergent since the Euclidean space is a Hilbert space. Let δPα=limn⟶∞δPn, we have δPα∈ℝm+1, by taking into account that Fα is continuous, we conclude that FαδPn⟶FαδPα, then, δPα=I. That proves the existence of a minimum of Fα.


## 2.5. Uniqueness Result

Lemma 4.
Fα admits a unique minimum δPα∈ℝm+1. This minimum δPα is the unique solution of the normal equation
(43)αδPα+HTHδPα=HTψ−U.Proof.
The question is to prove the uniqueness of the solution for the normal equation, because we have an equivalence between (27) and (28). We observe that the normal will be written as
(44)αI+HTHδPα=HTψ−U.
In order to show that this equation admits a unique solution, we must prove that the matrixαI+HTH is positive definite. Indeed, ∀δP≠0,(45)<αI+HTHδP,δP>=<αδP,δP>+<HTHδP,δP>+αδP22+HδP22.
Hence(46)<αI+HTHδP,δP>>0.
This proves that the matrix is positive definite, and so invertible, we deduce that equation (28) admits a unique solution, and therefore, Fα admits a unique minimum. Thus, a perturbation can be determined by (28)
(47)δPα=αI+HTH−1HTΨ−U.
Therefore, the solution can be characterized by (12) with a sufficiently large iteration numbers, or when the perturbation verified the convergence condition
(48)δPα2≤ε,for a given convergence precision ε.


## 3. Inversion Algorithm

Step 1.
Given the initial iterationP, the digital differential step τ, and the convergence precision ε, it is the additional condition ψt.Step 2.
Solve the direct problem using an implicit finite difference scheme featured in [16], get ul,tn;P, and ul,tn;p0,⋯,pi+τ,⋯,pm, for n=1,2,⋯, and i=0,1,⋯,m, then obtain the vector U, and the matrix H, by the formula (25).Step 3.
Choose an appropriate regularization parameterα≥0, and obtain a perturbation vector δPα, using formula (47), then obtain δDα.Step 4.
If there isδPα2≤ε, then the IA is finished, and D+δDα is considered the solution we just want to determine; otherwise, go to Step 2, replacing P with P+δP.


## 4. Numerical Simulations

In this section, we getT=1,l=1, the initial condition u0x=x, the number of spatial grids M=20, and the number of time grids N=20, and we propose the following two examples of simulations.


### 4.1. The First Case

For example, we chooseDx=2+x,0≤x≤1, as the exact source term. We performed the IA using the formula (47). By taking the initial data P0=0,0.65, the digital differential step τ=0.4, in the first case, we will do the simulation without the regularization term i.e., α=0, the computational results are listed in Table 1, where denotes fractional order, Dinv denotes inversion.Table 1
Effects of fractional order onD.
γDinvErr0.55(1.00000, 1.00000)1.2972e−70.6(1.00000, 1.00000)1.292844e−70.65(1.00000, 1.00000)5.54255e−80.7(1.00000, 1.00000)6.28358e−5Where Err given by Err=Dtrue−Dinv defines the absolute error in the solutions. The source term and their inversion are shown in Figure 1.Figure 1
Test without regularization forγ=0.65.Figure2 gives the graph of the error during iterations. Now, we put α=1e−6, and thus, we get Figure 3.Figure 2
Variation of the error function Err forγ=0.65.Figure 3
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7Figure4 gives the graph of the error during iterations.Figure 4
Variation of the error function Err forγ=0.65.


### 4.2. The Second Case

In the following, we use explicit regularization in algorithm execution due to the shortcomings of the problems being considered. Use this in.(49)Dx=1+x2,0≤x≤1,as the exact source term. We performed the IA using the formula (47). Considering the regularization parameter α=1e−9, the initial data P0=3.4,0,1.12, the digital differential step τ=0.5, and the convergent precision ε=1e−9, hence, we get the Table 2.Table 2
Effects of fractional order onD.
γDinvErr0.551.00000,1.00000,−3.874254e−95.874788e−70.61.00000,1.00000,−5.875477e−91.02157e−90.651.00000,1.00000,−7.700214e−85.14785e−80.71.00000,1.00000,−6.375155e−61.008547e−5Figure4 gives the graph of the error during iterations.The source term and their inversion are shown in Figures5(a) and 5(b).Figure 5
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7


### 4.3. The Third Case

This part considers the algorithm and we want to reconstruct the trigonometric(50)Dx=1+sin4πx,0≤x≤1,Table3 shows that the fractional order has a significant effect on the problem and the resulting algorithm. Solution errors reach their minimum in optimal regularization of parameters which correspond to each fractional order while regulating solutions. The regularization parameter is larger when the fractional order is approaching 1. The true source term and the inversion solution are presented by the Figures 6(a) and 6(b).Table 3
Effects of fractional order on the algorithm.
γαDinvErr0.553.212157e−91.00000,1.00000,1.500001.20025e−70.66.4757e−101.00000,1.00000,1.5000012.30215e−70.657.0002e−81.00000,1.00000,1.500001.385470e−60.76.15477e−111.00000,1.00000,1.500008.00145e−9Figure 6
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7The simulation is concerned by proposing a numerical solution of a one-dimensional fractional system with Caputo derivative. We use an extension of the well-known in literature by inverse problem method. The comparison between the exact solution and the numerical one of the considered system show the effectiveness of our proposed algorithm. The three cases that have been suggested demonstrate the veracity of our findings.


## 4.1. The First Case

For example, we chooseDx=2+x,0≤x≤1, as the exact source term. We performed the IA using the formula (47). By taking the initial data P0=0,0.65, the digital differential step τ=0.4, in the first case, we will do the simulation without the regularization term i.e., α=0, the computational results are listed in Table 1, where denotes fractional order, Dinv denotes inversion.Table 1
Effects of fractional order onD.
γDinvErr0.55(1.00000, 1.00000)1.2972e−70.6(1.00000, 1.00000)1.292844e−70.65(1.00000, 1.00000)5.54255e−80.7(1.00000, 1.00000)6.28358e−5Where Err given by Err=Dtrue−Dinv defines the absolute error in the solutions. The source term and their inversion are shown in Figure 1.Figure 1
Test without regularization forγ=0.65.Figure2 gives the graph of the error during iterations. Now, we put α=1e−6, and thus, we get Figure 3.Figure 2
Variation of the error function Err forγ=0.65.Figure 3
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7Figure4 gives the graph of the error during iterations.Figure 4
Variation of the error function Err forγ=0.65.


## 4.2. The Second Case

In the following, we use explicit regularization in algorithm execution due to the shortcomings of the problems being considered. Use this in.(49)Dx=1+x2,0≤x≤1,as the exact source term. We performed the IA using the formula (47). Considering the regularization parameter α=1e−9, the initial data P0=3.4,0,1.12, the digital differential step τ=0.5, and the convergent precision ε=1e−9, hence, we get the Table 2.Table 2
Effects of fractional order onD.
γDinvErr0.551.00000,1.00000,−3.874254e−95.874788e−70.61.00000,1.00000,−5.875477e−91.02157e−90.651.00000,1.00000,−7.700214e−85.14785e−80.71.00000,1.00000,−6.375155e−61.008547e−5Figure4 gives the graph of the error during iterations.The source term and their inversion are shown in Figures5(a) and 5(b).Figure 5
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7


## 4.3. The Third Case

This part considers the algorithm and we want to reconstruct the trigonometric(50)Dx=1+sin4πx,0≤x≤1,Table3 shows that the fractional order has a significant effect on the problem and the resulting algorithm. Solution errors reach their minimum in optimal regularization of parameters which correspond to each fractional order while regulating solutions. The regularization parameter is larger when the fractional order is approaching 1. The true source term and the inversion solution are presented by the Figures 6(a) and 6(b).Table 3
Effects of fractional order on the algorithm.
γαDinvErr0.553.212157e−91.00000,1.00000,1.500001.20025e−70.66.4757e−101.00000,1.00000,1.5000012.30215e−70.657.0002e−81.00000,1.00000,1.500001.385470e−60.76.15477e−111.00000,1.00000,1.500008.00145e−9Figure 6
Reconstruction result ofDtrue and Dinv.
(a)
γ=0.6(b)
γ=0.7The simulation is concerned by proposing a numerical solution of a one-dimensional fractional system with Caputo derivative. We use an extension of the well-known in literature by inverse problem method. The comparison between the exact solution and the numerical one of the considered system show the effectiveness of our proposed algorithm. The three cases that have been suggested demonstrate the veracity of our findings.


## 5. Conclusion

Inverse source problem for one dimensional time-space fractional diffusion equations (FDE) are considered. We use numerical approach determine the source term for a Caputo temporal. According to Hadamard sense, these inverse problem is ill-posed. In order to recreate a stable solution for the inverse issue, a new fractional Tikhonov regularization technique is applied. We conclude that the regularization coefficient and digital differential step show major effects on the algorithm, but for the first iteration with convergent precision, the influence on the algorithm is less. It also results from the conclusion that the first iteration with convergent precision has a small effect on the algorithm. Further research in the future can envisaged, and we cite the extension of the proposed method to higher dimensional space, also its application to solve optimal control problems governed by FDE.



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