# On the Recovery of a Conformable Time-Dependent Inverse Coefficient Problem for Diffusion Equation of Periodic Constraints Type and Integral Over-Posed Data

Authors: Mohammad Abdel Aal; Smina Djennadi; Omar Abu Arqub; Hamed Alsulami
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In the utilized analysis, we consider the inverse coefficient problem of recovering the time-dependent diffusion coefficient along the solution of the conformable time-diffusion equation subject to periodic boundary conditions and an integral over-posed data. Along with this, the conformable time derivative with order0<η≤1 is defined in the sense of a limit operator. The formal solution set for the considered inverse coefficient conformable problem is acquired via utilizing the Fourier expansion method. Under some conditions on the data and applicability of the Banach theorem, we insured the existence and uniqueness of the regular solution. Continuous dependence of the solutions set qt,ux,t in the given data is shown. Couples of illustrative examples in the form of data results and computational figures are also utilized. Future remarks, highlights, and work results are epitomized in the penultimate part. Finally, some latest used and focused references are given.


---

## Body

## 1. Prefatory Introduction

In all branches of applied mathematics, a forward problem is a problem of modeling a few physical fields, phenomena, or processes. The goal of solving a forward problem is to derive a function that describes its physical process. During the last decades, the mathematical construction based on the inversion of measurements which is named an inverse problem has been growing in interest. These problems form a multidisciplinary area joining applications of mathematics with many branches of sciences. For example, here, we try to list it briefly so that we do not prolong the reader and do not increase the size of the paper as much as possible, so the reader can refer to the references mentioned in this article to discover more. The authors of [1, 2] have discussed the applicability of the ICP in the fractional diffusion area with several theoretical results. The authors of [3] have determined the lost source term coefficients in the inverse DFM. The authors of [4] have studied the effect of the inverse Sturm-Liouville fractional problems. The authors of [5] have utilized a complete study on the final overdetermination for the inverse DFM.Many cosmopolitan researchers are interested in the inverse problem for DFM, integrodifferential equations, and heat equations where the time- or space-fractional derivatives are Riemann, Caputo, Fabrizio, tempered Caputo, or Atangana-Baleanu approaches as follows. The authors of [6] have utilized several inverse integrodifferential equations that involved two arbitrary kernels applying the Caputo fractional tempered derivative. The authors of [7] have discussed the ICP for DFM with nonlocal BCs. The authors of [8] have utilized the ICP for a multiterm DFM with nonhomogeneous BCs. The authors of [9] have proved the stability analysis and regularization in the ICP for DFM. The authors of [10] have discussed the uniqueness of the ICP for a multidimensional DFM. The authors of [11] have examined the inverse heat equation in a linear case that involved the Riemann fractional derivative. The authors of [12] presented the inverse DFM equation in its linear version that involved the Fabrizio fractional derivative together with an application on the Sturm-Liouville operator. Such derivative approaches can formulate various physical phenomena, for instance, Schrodinger equation [13, 14], delay differential models [15], telegraph equation [16], heat and fluid flows model [17], Neumann DFM [18].Conformable calculus proposed by [19] and generalized by [20] appears in various areas of applied sciences, abstract analysis, control, engineering, and biology as stellar mathematical agents to characterize the memory and hereditary behaviors of many processes and substances. It has been successfully applied in various areas of science and engineering (here, we try to list it briefly so that we do not prolong the reader and do not increase the size of the paper as much as possible, so the reader can refer to the references mentioned in this article to discover more) as in Newton mechanics [21], in solution of Burgers’ model [22], in time scales control problem [23], and in traveling wave field [24].It was applied to modeled diverse nonlinear conformable time-partial differential equation models with priority given to providing a more comprehensive explanation of chaos, dynamic systems, and the pattern of state change over time. Today, the notion of the CTD is one of the significant tools that appear in applied mathematics due to its suitability for the modulation of numerous real-world problems than the vintage derivative. Thereafter, the employ of the CTD has acquired remarkable refinement and awareness in many sections of engineering and theoretical sciences.Parameter identification shape OPD plays an important role in applied mathematics, engineering, and physics. The problem of recovering the diffusivity was studied by many researchers as follows. The authors of [25] have determined unknown source coefficients in the (space-time) DFM. The authors of [26] have exercised the quasi-boundary method for ICP related to the DFM. The authors of [27] have utilized the ICP related to the degenerate parabolic model in L2-space. The authors of [28] have presented several theoretical and experimental results of the DFM. The authors of [29] have tested the variational methods in the case of ICP for the Sturm-Liouville fractional problem. This work contributes to giving a solution set of an ICP for DFM involving CTD from an integral OPD specified condition together with periodic BCs. Anyhow, in the rectangle D=0,1×0,T, let us consider the general one-dimensional CTDE given by the subsequent formulation(1)Ttηux,t=qtuxxx,t+fx,t,and subject to the subsequent constraints(2)u0,t=u1,t,ux1,t=0,ux,0=ϕx,where x,t∈D, t∈0,T, x∈0,1, f∈CD⟶ℝ, ϕ∈C0,1⟶ℝ, and q∈C0,T⟶ℝ. Hither, Ttη stands for the CTD with order η∈0,1, qt is the diffusion coefficient, and fx,t is the source term map.As an upshot, if all functionsfx,t, qt, and ϕt are known, then the ICP (1) and (2) are mentioned as a direct problem. Anyhow, the ICP utilized here is to determine the coefficient diffusion term qt in (1) and (2) from an integral OPD condition given by(3)ktu0,t+∫01ux,tdx=Et,with kt=θ+βq−γt where β,γ,θ>0 are coefficients and are considered for unique solvability of the ICP, and E is a fully continuous map. The veracity of this type of ICP arises in the mathematical figuration of the technological process of external guttering applied [32, 33]., the CTD and the conformable integral with orderη∈0,1are as(4)Ttηux,t=limε⟶0ux,t+εt1−η−ux,tε,ℐ0ηux,t=∫0tux,ξξ1−ηdξ.After the formulation of the problem and the formation of some basic related results in the prefatory introduction, the rest of the utilized analysis is epitomized as follows:(i)
Phase 1: some requisite results related to the spectral problem including the eigenfunctions and eigenvalues of the spectral and its conjugate problem are recalled in section2(ii)
Phase 2: unique existence of a regular solution is proved in section3(iii)
Phase 3: continuous dependence of the solution on the given data is proved in section4(iv)
Phase 4: illustrative application examples are utilized in section5(v)
Phase 5: work results, highlights, and future work are presented in section6


## 2. Spectral Problem and Series Representation

This section is intended to expand the solution of an ICP for the CTDE insight of BCs and an integral OPD constraint. The solution-based approach and its theoretical concept are derived with consistency from the FEM.Let us fundamentally consider the subsequent spectral problem on0≤x≤1:(5)S″x+λSx=0,S0=S1,S′1=0.Certainty, problem (5) is well-known in [36] as the auxiliary spectral problem for solving a boundary value problem. Anyhow, (5) has the eigenvalues and the corresponding eigenfunctions as(6)λn=2nπ2,n=0,1,2,…,(7)S0x=2,S2n−1x=4cos2πnx,n=1,2,…,S2nx=41−xsin2πnx,n=1,2,….The set of eigenfunctionsSnn=1∞ forms a basis for L20,1 space, and the adjoint problem to (5) has the subsequent form on 0≤x≤1(8)Z″x+λZx=0,Z0=0,Z′0=Z′1=0.The system of eigenfunctions and associated functions of(8)is denoted by(9)Z0x=x,Z2n−1x=xcos2πnx,n=1,2,…,Z2nx=sin2πnx,n=1,2,….As an upshot, (6) and (7), and (9) form a biorthonormal system on 0,1. The coming lemmas are important for the mathematical analysis of the ICP.Lemma 1.
Ifg∈C30,1 satisfies the constraints g0=g1, g′1=0, and g″1=g″0, then(10)∑n=1∞λng2n≤c1gC3,∑n=1∞λng2n−1≤c2gC3,for some constants c1 and c2 with gn=∫01gxZnxdx.Proof.
The subsequent equality(11)λng2n=−1λn∫01g″xcosλnxdx,
holds by applying three times parts integrations and that toog0=g1 and g″1=g″0. Indeed, the equality(12)λng2n−1=−1λn∫01xg″x+2g′xcosλnxdx,
holds by applying two times parts integrations and that toog0=g1 and g′1=0. Using the Bessel and Schwarz inequalities for (11) and (12), we get (10).Lemma 2.
Ifqjt∈C0,T satisfies the constraints 0<a≤qj with j=1,2, then for n∈ℕ and ∀t∈0,T, we take out the inequality(13)e−λn∫0tsη−1q1sds−e−λn∫0tsη−1q2sds≤1aeq1−q2C.Proof.
By utilizing the mean value theorem results one−x, one gained ∃ϑ with λn∫0tsη−1q1sds≤ϑ≤λn∫0tsη−1q2sds such that(14)e−n∫0tsη−1q1sds−e−n∫0tsη−1q2sds=e−ϑλn∫0tsη−1q1s−q2sds≤λntηηe−λnatη/ηq1−q2C.
At last, by mentionxe−bx≤1/be with x≥0 and b=cst>0, we obtain the (13).


## 3. Existence-Uniqueness of Solution

This section is intended to justify the existence of the classical solution setqt,ux,t that is to show under constraints (2) and (3), and qt>0 that qt∈C0,T and ux,t∈C2,1D∩C1,0D. The derivation-based approach depends on the contraction approach and the Banach fixed point theorem.First, according to our classical technique of the FEM, we determine the solutionux,t of (1) and (2) as in the subsequent form(15)ux,t=ϕ0+∫0tsη−1f0sdsS0x+∑n=1∞ϕ2ne−λn∫0tsη−1qsds+∫0tsη−1f2nse−λn∫styη−1qydydsS2nx+∑n=1∞ϕ2n−1−4πnϕ2nte−λn∫0tsη−1qsds+∫0tf2n−1s−4πnf2nst−ssη−1e−λn∫styη−1qydydsS2n−1x,where the coefficient ϕn and the function fnt are given as(16)ϕn=∫01ϕxZnxdx,fnt=∫01fx,tZnxdx.Now, let us drive the solution of the ICP from the OPD specified condition (3) as in the next assumptions(17)Fkt=kt,(18)Fkt=1Φt2ϕ0+∫0tsη−1f0sds+2π∑n=1∞1nϕ2ne−λn∫0tsη−1qsds+∫0tsη−1f2nse−λn∫styη−1qydyds−Et,(19)Φnt=−2ϕ0+∫0tsη−1f0sds+4∑n=1∞4πntϕ2n−ϕ2n−1e−λn∫0tsη−1qsds+∫0t4πnf2nst−s−f2n−1ssη−1e−λn∫styη−1qydyds,(20)qt=βkt−θ1/γ.Theorem 1.
Under the subsequent constraints,(1)
ϕ∈C30,1 such that(i)
ϕ0=ϕ1, ϕ′1=0, and ϕ″1=ϕ″0(ii)
ϕ2n≥0 and ϕ2n−1≤0 with n=1,2,… and ϕ0+2ϕ1<0(2)
f∈CD¯ and f∈C30,1 for arbitrarily fixed t∈0,T such that(i)
f0,t=f1,t, fx1,t=0, and fxx0,t=fxx1,t(ii)
f2nt≥0 and f2n−1t≤0 with n=1,2,… and f0t+2f1t<0(3)
Et∈C0,T and satisfies Et<2ϕ0+f0t, ∀t∈0,T
There exist positive numbersθ0 and γ0 such that the ICP (1) and (2), and (3) with the parameters θ<θ0 and γ<γ0 has a unique solution.Proof.
First, the sums involved inux,t and uxx,t are continuous in D¯, since by using the results of Lemma 2; the series (16) and its x-partial derivative are uniformly convergent in D¯, next, since the series(21)∑n=1∞λn3e−Cλnϵ,is convergent. Then, the CTD and the second-order derivative of the series ux,t concerning x are uniformly convergent for t≥ϵ>0. So(22)ux,t∈C2,1D∩C1,0D¯.
Now, let us consider(23)E0=2mint∫0tsη−1f0sds−maxtEt,E1=2maxt∫0tsη−1f0sds+maxt∑n=1∞2nπ∫0tsη−1f2nsds−mintEt,
together with(24)θ0=1Ψn2ϕ0+E0,θ1=1−4ϕ1+∫0Tsη−1f1sds−2ϕ0+∫0Tsη−1f0sds2ϕ0+2π∑n=1∞1nϕ2n+E1,Ψn=−2ϕ0+∫0Tsη−1f0sds+4∑n=1∞4πnTϕ2n−ϕ2n−1+∫0T4πnT−sf2ns−f2n−1ssη−1ds.
Then, using(18)and(21)over0≤t≤T, one gained(25)0<θ0≤kt≤θ1.
Under the conditionθ0>θ, the subsequent inequalities are hold(26)0<βθ1−θ1/γ≤qt≤βθ0−θ1/γ.
Now, let us show thatF:Cθ0,T⟶Cθ0,T and that F is a contraction mapping in Cθ0,T for small θ and large γ, where Cθ0,T=kt∈C0,T:θ0≤kt≤θ1,∀t∈0,T. Anyhow, let k1t,k2t∈Cθ0,T, we have(27)Fk1t−Fk2t=1−2ϕ0+∫0tsη−1f0sds+θ12t⋅θ01t−Et+2ϕ0+∫0tsη−1f0sds−2ϕ0+∫0tsη−1f0sds+θ11tθ12t−θ11t−θ02t−θ01t,
withθ0mt, θ0mt, and qmt for m=0,1 are given as subsequent(28)θ0mt=2π∑n=1∞1nϕ2ne−λn∫0tsη−1qmsds+∫0tsη−1f2nte−λn∫styη−1qmydyds,θ1mt=4∑n=1∞4πnϕ2nt−ϕ2n−1e−λn∫0tsη−1qmsds+∫0t4πnf2nst−s−f2n−1sη−1e−λn∫styη−1qmydyds,qmt=βkmt−θ1/γ.
Based on Lemma2 and inequality (26), one can collect(29)2π∑n=1∞1nϕ2ne−λn∫0tsη−1q2sds−e−λn∫0tsη−1q1sds+∫0tsη−1f2nse−λn∫styη−1q2ydy−e−λn∫styη−1q1ydyds≤2πe∑n=1∞1nϕ2n+∫0tsη−1f2nsdsβθ1−θ−1/γq2−q1C0,T,(30)θ12t−θ11t=4∑n=1∞4πnϕ2nT−ϕ2ne−λn∫0tsη−1q2sds−e−λn∫0tsη−1q1sds+∫0T4πnf2nsT−s−f2n−1sη−1e−λn∫styη−1q2ydy−e−λn∫styη−1q1ydyds≤4e∑n=1∞4πnϕ2nT−ϕ2n+∫0Tf2n−1s−4πnf2nsT−ssη−1ds⋅βθ1−θ−1/γq2−q1C.
Putting (29) and (30) into (27), we get(31)max0≤t≤TFk1t−Fk2t≤βθ1−θ−1/γlq2−q1C,(32)l=−2πe14ϕ1+∫0Tsη−1f1sds+2ϕ0+∫0Tsη−1f0sds−2πθ1∑n=1∞ϕ2n−1−4πnϕ2nT+∫0Tf2n−1s−4πnf2nsT−ssη−1ds+∑n=1∞nϕ2n+∫0Tsη−1f22sds.
By using the mean value theorem and (26), we show that(33)q2t−q1t≤β1/γγθ0−θ1+1/γk2t−k1t.
From (31) and (33), we deduce that(34)Fk2−Fk1C≤lθ0−θγθ1−θθ0−θ1/γk2−k1C.
We fix a sufficiently large numberγ0>0 such that(35)M=lθ0−θγ0θ1−θθ0−θ1/γ≤1.
In the caseγ>γ0 and according to the Banach fixed point theorem, we obtained that (19) has a unique solution qt∈Cθ0,T.


## 4. Continuously Dependent on the Data

This section is focused on the continuous dependence of the solutions set on the given data. In other words, some stability analysis is derived from the ICP (1) and (2), and (3) insight of CTD.First, considering a solution setqt,ux,t, where’s ux,t, uxxx,t, and Ttηux,t are in C0,1×0,1⟶ℝ and q∈C0,T⟶ℝ.Theorem 2.
Consider the given data in the formfx,t,ϕx,Et which satisfies the assumptions (1), (2), and (3) of Theorem 1. Assume that(36)2ϕ0+E0≥M0>0,ϕ0+2ϕ1≤−M1,f0t+2f1t≤−M2.
together with(37)EC≤M3,ϕC3≤M4,fC3,1≤M5,
for someMi>0 with i=0,…,5. So, the set qt,ux,t of ICP (1), (2), and (3) depends continuously on the given data fx,t,ϕx,Et.Proof.
Letux,t,qt and u˜x,t,q˜t be two solution sets of (1), (2), and (3) relating to the data sets f,ϕ,E and f˜,ϕ˜,E˜, simultaneously. Utilizing (18), one has(38)kt=1Φnt2ϕ0+∫0tsη−1f0sds+2π∑n=1∞1nϕ2ne−λn∫otsη−1qsds+∫0tsη−1f2nse−λn∫styη−1qydyds−Et,qt=βkt−θ1/γ,k˜t=1Φ˜nt2ϕ˜0+∫0tsη−1f˜0sds+2π∑n=1∞1nϕ˜2ne−λn∫0tsη−1q˜sds+∫0tsη−1f˜2nse−λn∫styη−1q˜ydyds−E˜t,q˜t=βk˜t−θ1/γ.
In which,Φnt and Φ˜nt are given as the subsequent(39)Φnt=−2ϕ0+∫0tsη−1f0sds+4∑n=1∞4πntϕ2n−ϕ2n−1e−λn∫0tsη−1qsds+∫0t4πnf2nst−s−f2n−1ssη−1e−λn∫styη−1qydyds,Φ˜nt=−2ϕ˜0+∫0tsη−1f˜0sds+4∑n=1∞4πnϕ˜2nt−ϕ˜2n−1e−λn∫0tsη−1q˜sds+∫0t4πnf˜2nst−s−f˜2n−1ssη−1e−λn∫styη−1q˜ydyds.
Herein, by using the Schwarz and the Bessel inequalities together with (10) and (13), it is easy to estimate the subsequent quantities(40)∑n=1∞1nϕ2ne−λn∫0tsη−1qsds+∫0tsη−1f2nse−λn∫styη−1qydyds≤CϕC3+fC3,1,∑n=1∞4πnϕ2nt−ϕ2n−1e−λn∫0tsη−1qsds+∫0t4πnf2nst−s−f2n−1ssη−1e−λn∫styη−1qydyds≤1+4πTCϕC3+fC3,1,∑n=1∞1nϕ2ne−λn∫0tsη−1qsds+∫0tsη−1f2nse−λn∫styη−1qydyds−∑n=1∞1nϕ˜2ne−λn∫0tsη−1q˜sds+∫0tsη−1f˜2nse−λn∫styη−1q˜ydyds≤N1ϕ−ϕ˜C3+N2q−q˜C3+N3f−f˜C3,∑n=1∞1nϕ2n−1−4πnϕ2nte−λn∫0tsη−1qsds+∫0tf2n−1s−4πnf2st−ssη−1e−λn∫styη−1qydyds−∑n=1∞1nϕ˜2n−1−4πnϕ˜2nte−λn∫0tsη−1q˜sds+∫0tf˜2n−1s−4πnf˜2nst−ssη−1e−λn∫styη−1q˜ydyds≤N4ϕ−ϕ˜C3+N5q−q˜C3+N6f−f˜C3,where Ni with i=1,2,…,6 are positive constants. Thus, one can write(41)kt−k˜t≤N7ϕ−ϕ˜C3+N8q−q˜C+N9f−f˜C3+N10E−E˜C,where Ni with i=7,8,9,10 are positive constants. Hence, from (32), we have for θ<θ0 that(42)qt−q˜t≤β1/γγθ0−θ1+1/γkt−k˜t,
withθ0≥2ϕ0+E0/N11fC3+ϕC3≥M0/N11M3+M4. If θ is sufficiently small such that θ<M0/N11M3+M4 by using (41) in (42), we get(43)kt−k˜tC≤Nϕ−ϕ˜C3+f−f˜C3+E−E˜C,
for some positive constantN. In the end, this proves that q continuously depends on the input data. Similarly, one can deal with the above results to rely that ux,t depends continuously upon the given data.


## 5. Illustrative Application Examples

Through this part, we are going to present some examples of the ICPs for the CTDE. Anyhow, to show the theoretical outcomes of the previous sections, we illustrate two application examples. That is, we will show through these two applications that the solutionqt,ux,t depends continuously on the order 0<η≤1 input data.The reader should remember that we used the MATHEMATICA 11 program in our calculations for the numerical tables and our drawings of figures.Example 1.
Consider the ICP (1), (2), and (3) for the CTDE in the domain 0,1×0,1 with the given data:(44)Ttηux,t=qtuxxx,t+cos2πe−tηη−1,
and subject to the subsequent constraints(45)u0,t=u1,t,ux1,t=0,ux,0=−cos2πx,ktu0,t+∫01ux,tdx=−1+4π2e−tη/ηe−tη/η,
withθ=β=γ=1 and kt=1+4π2e−tη/η.
Simple manipulations yield that the analytical solutions setqt,ux,t can be formed as(46)qt=14π2etη/η,ux,t=−cos2πxe−tη/η.Example 2.
Consider the ICP (1), (2), and (3) for the CTDE in the domain 0,1×0,1 with the given data:(47)Ttηux,t=qtuxxx,t+1−xsin2πx1−ηe−tη,
and subject to the subsequent constraints(48)u0,t=u1,t,ux1,t=0,ux,0=1−xsin2πx,ktu0,t+∫01ux,tdx=14π2e−tη,with θ=β=γ=1 and kt=1+4π2e−tη.
Simple manipulations yield the analytical solutions setqt,ux,t that can be formed as(49)qt=14π2etη,ux,t=1−xsin2πxe−tη.
Next, some computational figures for the analytical solution setqt,ux,t are analyzed and utilized in the form of one- and two-dimensional plots towards the continuous behavior over the order η∈0,1. Anyhow, Figures 1 and 2 illustrate the solution ux,t in the case of η∈1,0.85,0.7,0.55 for examples 1 and 2, simultaneously whilst Figure 3 illustrates the solution qt in the case of η∈1,0.85,0.7,0.55 for examples 1 and 2 together.
Right after that, as an important application result, from Figures1, 2, and 3, we show graphically that any small change in the input order η∈0,1 leads to a change in the solution.
Ultimately, some computational data for the analytical solution setqt,ux,t are analyzed and utilized in the form of tables towards the continuous behavior over the order η∈0,1. Anyhow, Tables 1 and 2 illustrate the solution set in the case of η∈1,0.85,0.7,0.55 for examples 1 and 2, simultaneously.
Right after that, as an important application result, from Tables1 and 2, we show tabularly that any small change in the input order η∈0,1 leads to a change in the solution.Figure 1
Plot of the 3D analytical solutionux,t in example 1 on 0,1×0,1: (a) η=1, (b) η=0.85, (c) η=0.7, and (d) η=0.55.
(a)(b)(c)(d)Figure 2
Plot of the 3D analytical solutionux,t in example 2 on 0,1×0,1: (a) η=1, (b) η=0.85, (c) η=0.7, and (d) η=0.55.
(a)(b)(c)(d)Figure 3
Plot of the 2D analytical solutionqt on 0,1 as Red: η=1, Green: η=0.85, Yellow: η=0.7, and Blue: η=0.55 in (a) example 1 and (b) example 2.
(a)(b)Table 1
Tabulated data of the analytical solution setqt,ux,t on 0,1×0,1 over the order η∈0,1 in example 1.
xtη=0.55η=0.7η=0.85η=1ux,t,qtux,t,qtux,t,qtux,t,qt0.20.2−0.146,0.054−0.194,0.040−0.229,0.034−0.253,0.0310.4−0.103,0.076−0.146,0.054−0.180,0.043−0.207,0.0380.6−0.078,0.100−0.144,0.069−0.144,0.054−0.170,0.0460.8−0.062,0.126−0.091,0.086−0.177,0.067−0.139,0.0560.40.20.382,0.0540.509,0.0400.600,0.0340.662,0.0310.40.270,0.0760.381,0.0540.471,0.0430.542,0.0380.60.205,0.1000.298,0.0690.378,0.0540.444,0.0460.80.162,0.1260.238,0.0860.306,0.0670.364,0.0560.60.20.382,0.054x0.509,0.0400.600,0.0340.662,0.0310.40.270,0.0760.381,0.0540.471,0.0430.542,0.0380.60.205,0.1000.298,0.0690.378,0.0540.444,0.0460.80.162,0.1260.238,0.0860.306,0.0670.364,0.0560.80.2−0.146,0.054−0.194,0.040−0.229,0.034−0.253,0.0310.4−0.103,0.076−0.146,0.054−0.180,0.043−0.207,0.0380.6−0.078,0.100−0.114,0.069−0.144,0.054−0.170,0.0460.8−0.062,0.126−0.091,0.086−0.117,0.067−0.139,0.056Table 2
Tabulated data of the analytical solution setqt,ux,t on 0,1×0,1 over the order η∈0,1 in example 2.
xtη=0.55η=0.7η=0.85η=1ux,t,qtux,t,qtux,t,qtux,t,qt0.20.20.504,0.0380.550,0.0350.590,0.0330.623,0.0310.40.416,0.0460.449,0.0430.481,0.0400.510,0.0380.40.358,0.0540.378,0.0510.398,0.0480.418,0.0460.80.314,0.0610.323,0.0600.333,0.0580.342,0.0560.40.20.233,0.0380.255,0.0350.273,0.0330.289,0.0310.40.193,0.0460.208,0.0430.223,0.0400.236,0.0380.60.166,0.0540.175,0.0510.185,0.0480.194,0.0460.80.146,0.0610.150,0.0600.154,0.0580.158,0.0560.60.2−0.156,0.038−0.170,0.035−0.182,0.033−0.192,0.0310.4−0.129,0.046−0.139,0.043−0.149,0.040−0.158,0.0380.6−0.110,0.054−0.117,0.051−0.123,0.048−0.129,0.0460.8−0.097,0.061−0.100,0.060−0.103,0.058−0.106,0.0560.80.2−0.126,0.038−0.138,0.035−0.147,0.033−0.156,0.0310.4−0.104.0.046−0.112,0.043−0.120,0.040−0.128,0.0380.6−0.089,0.054−0.095,0.051−0.100,0.048−0.104,0.0460.8−0.079,0.061−0.081,0.060−0.083,0.058−0.085,0.056


## 6. Work Results, Highlights, and Future Work

The ICP which involves determining the time-dependent coefficient for the CTDE has been investigated and utilized successfully in this research analysis in the form of theoretical and practical. We have used the eigenfunctions of spectral and adjoint problem approach to writing an explicit solution of the direct problem, and then, we have used the over-posed data to derive the solution of the presented ICP. The existence and uniqueness results of identifying the time-dependent coefficient are formatted and proved by using the Banach fixed point theorem. Also, the continuous dependence upon the given data is proved too. Couples of illustrative examples are utilized, discussed, and shown in the form of data results and computational figures. Our future work will focus on the similar utilized analysis insight of the fractional M-time derivative approach.



---

*Source: 5104725-2022-10-21.xml*