Inverse source problem for the pseudoparabolic equation associated with the Jacobi operator

In this paper, we investigate direct and inverse problems for time-fractional pseudoparabolic equations associated with the Jacobi operator. The existence and uniqueness of the solutions are proven. Also, the stability result of the inverse source problem (ISP) is established

For the ISP we will restore the pair (u, f ) under some conditions on the function ψ.
One of the first mathematicians who studied the ISP was Rundell [Run80] in 1980s.He considered the evolution type equation (1.2) du dt + Au = f in a Banach space X, where A is linear operator in X and f is a constant vector in X, with conditions u(0) = u 0 , and u(T ) = u 1 .Using semigroups of operators Rundell proved a general theorem about the existence of a unique solution pair (u(t), f ) of the problem, which then was applied to equations of parabolic and pseudo-parabolic types.When the non-homogeneous term is represented in the form f (t) = Φ(t)f , where Φ(t) is known operator and the element f is unknown, and A is a closed linear operator from L p (Ω) into L p (Ω), several ISPs for the equation (1.2) were studied by A.I. Prilepko and I.V. Tikhonov [PT92] in 1992.They applied obtained results to the transport equation.In the general case, where the unknown source depends on time, under a sufficient condition, ISPs for the equation (1.2) with the linear elliptic partial differential operator A of order 2m with the bounded measurable coefficients such that (Aϕ, ϕ) ≥ ϕ 2 for all ϕ ∈ H 2m (Ω) ∩ H m 0 (Ω), µ = constant > 0 was investigated by I. Bushuyev [Bus95] in 1995.
Nonetheless, there is no general closed theory for abstract case of F (x, t).Known results deal with separated source terms.In 2002 I.V. Tikhonov and Yu.S. Eidelman [TE02] considered ISPs for the generalization of the equation (1.2) of the form dt N = Au(t) + p, 0 < t < T, for some positive integer N ≥ 1 and some real number T > 0 with an unknown parameter p and a closed linear operator A in the Banach space under the Cauchy conditions and "over-determination condition" u(T ) = u N (also in the Banach space).
For the Laplace operator (−∆) which is one of the most interesting examples in Physics, M. Choulli and M. Yamamoto in [CY04] established the uniqueness and conditional stability in determining a heat source term from boundary measurements with f = σ(t)ϕ(x), where σ(t) is known.
Asymptotic behaviour of the solution of the inverse source problem for the pseudoparabolic equation with a integral over-determination condition was studied by M. Yaman and Ö. F. Gözükızıl in [YG03] in 2004.
Fractional derivatives and fractional partial differential equations have received great attention both in analysis and application, which are used in modeling several phenomena in different areas of science such as biology, physics, and chemistry, so the fractional computation is increasingly attracted to mathematicians in the last several decades.ISP for the time fractional parabolic equation where c D α t is the Caputo derivative defined by and L is a symmetric uniformly elliptic operator was considered by K. Sakamoto and M. Yamamoto in [SY11] in 2011.The authors proved that the inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants using final overdetermining data.Blow-up solution and stability to ISP for the pseudo-parabolic equation with the integral overdetermination condition was studied by Metin Yaman in [Yam12] in 2012.ISP for the equation (1.2) considered by M.M. Slodicka in [Slo13] in 2013, when A is a linear differential operator of second-order, strongly elliptic, and the righthand side f is assumed to be separable in both variables x and t, i.e. f (x, t) = g(x)h(t) (in this case h(t) is unknown).ISP for a semilinear time-fractional diffusion equation of second order in a bounded domain in R d with a linear second order differential operator L(x, t) in the divergence form with space and time dependent coefficients was studied by M. Slodicka and M. Siskova in [SS16] in 2016.Authors showed the existence, uniqueness and regularity of a weak solution (u, h) ([SS16, Theorem 2.1, p. 1658]).One of the recent papers for inverse source problems for pseudo-parabolic equations with fractional derivatives is [RSTT21] (in 2021).In [RSTT21], authors have considered solvability of an inverse source problem for the pseudo-parabolic equation with the Caputo fractional derivative D α t of order 0 < α ≤ 1 , where H be a separable Hilbert space and L, M be operators with the corresponding discrete spectra on H.The authors obtained well-posedness results.
The semigroups (H ) t≥0 (the solution of the heat equation associated with the Jacobi-Dunkl operator Λ 2 α,β ) generate a new family of Markov processes on the real line.On some Riemannian symmetric spaces this process is the radial part of the Brownian motion for particular values of (α, β) [CGM06].
However, the ISP for the pseudo-parabolic equations generated by the Jacobi operator ∆ α,β (1.1) have not been still considered.So, our goal is to consider the ISP for the pseudo-parabolic equation with this special operator.Harmonic analysis associated with the operator ∆ α,β has been studied by M. Flensted-Jensen and T. H. Koornwinder [FJ72,FJK73,FJK79,Koo75].The spectral decomposition of the Jacobi operator was considered by M. Flensted-Jensen in 1972 [FJ72].There were obtained a generalization of the classical Paley-Wiener Theorem and a generalized Fourier transform F α,β , is called Jacobi-Fourier transform.Eigenfunctions ϕ α,β λ (x) of the operator Jacobi is called the Jacobi function, which is hypergeometric function.The pseudo-differential operators (see [SD98]) and Sobolev type spaces G s,p α,β (see [SD00]) associated with the Jacobi operator was studied by N. Ben Salem and A. Dachraoui.In [SD98], authors proved that a pseudo-differential operator associated with a symbol in S m 0 is a continuous linear mapping from some subspace of the Schwartz space into itself.
Our main result reads as follows.
Theorem 1.1.Let 0 < γ ≤ 1. Assume that ψ, φ ∈ H. Then the pair (u, f ) is a unique solution of the ISP, which are functions . The contents of this paper as follows.In Section 2, we collect some results about harmonic analysis associated with the Jacobi operator on R + and here we introduce the Sobolev type space H, also given some necessary information about fractional derivative.In Section 3, we prove Theorem 3.1 for the direct problem.In Section 4, we prove our main Theorem 3.2 about solvability of the inverse source problem associated with the Jacobi operator on R + , also shown stability analysis and example for the inverse source problem.
Let S e (R) be the space of even, rapidly decreasing, and C ∞ -functions on R, equipped with usual Schwartz topology, and S r e (R) = {(cosh x) r S e (R)}, 0 < r ≤ 2 be the space with the topology defined by the semi-norms Clearly S r e (R) is invariant under ∆ α,β and the semi-norms defined by Let L p (R + , ν α,β ), 1 ≤ p < ∞ be the space of measurable functions g on R + such that Here, c α,β (λ) is the Harish-Chandra function, given by For short, we use notations L p (µ) and L p (ν) instead L p (R + , µ α,β ) and L p (R + , ν α,β ), respectively. For and for g ∈ L 1 (ν) the inverse Fourier-Jacobi transform F −1 α,β is given by where ϕ α,β λ is the Jacobi functions (2.1).

Main Results
In this Section we deal with the direct problem for the time-fractional pseudoparabolic equation associated with the Jacobi operator ∆ α,β (1.1).Moreover, ISPs are subject to study.The existence, uniqueness and stability results are established.
3.1.The direct problem for the time-fractional pseudo-parabolic equation with the Jacobi operator.Let 0 < γ ≤ 1.We consider the non-homogeneous time-fractional pseudo-parabolic equation where the functions f and φ are given functions.Our aim is to find unique solution u of the problem (3.1) -(3.2).H) and can be represented by formula Proof.We assume that 0 < γ ≤ 1, λ ∈ R + and u(t, •) ∈ H.We first prove that the problem (3.1)-(3.2) has only one solution, if the later exists.Suppose the proposition were false.Assume that there exist two different solutions u 1 (t, x) and u 2 (t, x).Denote u 0 (t, x) = u 1 (t, x) − u 2 (t, x).Then u 0 (t, x) solves the following equation The problem (3.3)-(3.4)has only trivial solution.This implies uniqueness of the solution.Now, we will prove the existence of the solutions.Using the Fourier-Jacobi transform F α,β (2.2) on both sides of (3.1)-(3.2),we have for all λ ∈ R + and 0 < t < T .The solution (see [KST06, p. 231, ex.4.9]) of the problem (3.5)-(3.6) is given by where E γ,1 (z) is the classical Mittag-Leffler function and E γ,γ (z) is the Mittag-Leffler type function.Now by using the inverse Fourier-Jacobi transform F −1 α,β (2.3) to (3.7), we obtain the formula for the solution of the problem (3.1)-(3.2),given by By using the property of the Mittag-Leffler function, we obtain and we can write (3.7) in a form by using the rule integration by parts and E γ,1 (0) = 1.Let 0 < γ < 1 and f ∈ C 1 ([0, T ], L 2 (µ)), φ ∈ H, then we can estimate u as follows here we have used Cauchy-Schwarz inequality, Fubibi's theorem and a b denotes a ≤ cb for some positive constant c independent of a and b.Thus, Then, we obtain H by using Cauchy-Schwarz inequality and Fubini's theorem.Thus, Then, we have Consequently, using Definition 2.10 we obtain u ∈ C γ ([0, T ], L 2 (µ)).Our prove is completed.

3.2.
The ISP for the time-fractional pseudo-parabolic equation.This subsection deals with the ISP for the time-fractional pseudo-parabolic equation associated with the Jacobi operator ∆ α,β (1.1).

3.2.1.
Statement of the problem.Let 0 < γ ≤ 1.We aim to find a couple of functions (u, f ) satisfying equation . Proof.We assume that 0 < γ ≤ 1, and u(t, •), f ∈ H. Let us first prove the existence result.By using the Fourier-Jacobi transform F α,β (2.2) on both sides of (3.8)-(3.10),we obtain Solution of the equation (3.11) is given by for all 0 < γ ≤ 1 and functions f (λ) and C(λ) are unknown functions.For determine these functions we use conditions (3.12) and (3.13).After that we have Thus we have Substituting the resulting functions C(λ) and f (λ) into (3.14),we get Therefore solution of the problem (3.11) -(3.13) is the pair ( u, f ).We obtain solution of the problem (3.8)-(3.10)by applying inverse Fourier-Jacobi transform F −1 α,β (2.3) to the functions u and f , i.e. (3.16) λ (y)ϕ α,β λ (x)dµ α,β (y)dν α,β (λ) and Let ψ, φ ∈ H. Then using Lemma 2.12 we can estimate the function u as following The existence is proved.Now, let us prove the uniqueness of the solution.Taking into account the property of the Fourier-Jacobi transform Proposition 2.4, one observes that a pair of functions (u, f ) is uniquely determined by the formulas (3.16) and (3.17).The uniqueness is proved.
Illustrations of our calculations above are given in Table 1.

Table 1 .
Stability Test Conclusion.Table 1 confirms that the solution of the problem (3.8)-(3.10) is continuously depending on the given data.Small changes in the given data imply small changes in (u, f ). 4. Appendix Calculations in Table 1 are made by using Maple 2021 program with the following codes: