Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation

Article History: Received 16 December 2017 Accepted 08 February 2018 Available 11 April 2018 On the basis of a reproducing kernel Hilbert space, reproducing kernel functions for solving the coefficient inverse problem for the kinetic equation are given in this paper. Reproducing kernel functions found in the reproducing kernel Hilbert space imply that they can be considered for solving such inverse problems. We obtain approximate solutions by reproducing kernel functions. We show our results by a table. We prove the efficiency of the reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation.


Introduction
Kinetic theory emerged with Maxwell and Boltzmann, Hilbert, Enskog, Chapman, Vlasov, and Grad.Investigating for a form of matter which could clarify Saturn's rings, Maxwell considered that they were performed of rocks colliding and gravitating around the planet.The density of matter is then parameterized by the space position x and the velocity v of the rocks.Boltzmann modeled the operation, endowed a common representation of a dilute gas as particles undergoing collisions and with free motion between collisions, and he found the famous equation which is now named after him [1].Vlasov obtained another kinetic equation (KE) for plasmas of charged particles.Kinetic equations (KEs) rise in a variety of sciences and implementations such as astrophysics, aerospace engineering, nuclear engineering, particle fluid interactions and semi-conductor technology recently.The general property of these models is that the underlying Partial Differential Equation is posed in the phase space (x, v) ∈ R n n ≥ 1, [2].
We consider the problem of obtaining (f, σ) in Ω from the following equation [1]: with the boundary conditions: In this work, the reproducing kernel functions for solving a coefficient inverse problem (IP) for the KE are given.Reproducing kernels were used for the first time at the beginning of the twentieth century by Zaremba in his work on boundary value problems for harmonic and biharmonic functions [3,4].The general theory of reproducing kernel Hilbert spaces was established simultaneously and independently by Aronszajn [5] and Bergman [6] [11].Wahba used regression design for some equivalence classes of kernels [12].Nashed et al. found regularization and approximation of linear operator equations in reproducing kernel spaces [13].Al e'damat applied analytical-numerical method for solving a class of two-point boundary value problems [14].For more details see [15][16][17][18][19][20][21].

Reproducing kernel functions
In this section, we give some important reproducing kernel functions.
Definition 1. Hilbert function space H is a reproducing kernel space if and only if for any fixed x ∈ X, the linear functional Definition 2. We describe the space T 2 2 [1, 2] as: The inner product and the norm in T 1 2 [1,2] are obtained as follow: Theorem 1. Reproducing kernel function A k of reproducing kernel space T 2 2 [1,2] is found as follow: where by integration by parts.Then, we get This completes the proof.Definition 3. We describe the space M 2 2 [−1, 1] by: The inner product and the norm in by integration by parts.Then, we get This completes the proof.
Definition 4. We describe the space M 1 2 [−1, 1] as: and ] is a reproducing kernel space, and its reproducing kernel function E k is given as [15]: Definition 5. We define the space The inner product and the norm in T 1 2 [1,2] are given as: and Lemma 2. The space T 1 2 [1, 2] is a reproducing kernel space, and its reproducing kernel function F p is given as [15]:

Main results
Definition 6.If m + n > 2, define the binary space [22] W (m,n) 2 (Ω) with the inner product We found the main reproducing kernel function for the problem in this section.We take 16 .
In our problem m = 2, n = 2.We obtain the main reproducing kernel function as: where, (Ω).

Applications
The solution of ( 1)-( 3) is given in the reproducing kernel space W (2,2) 2 (Ω) in this section.On defining the linear operator N : (Ω), after homogenizing the boundary conditions, model problem (1)-( 3) changes to the problem Lemma 3. N is a bounded linear operator.
Therefore, we have a Thus, we get This completes the proof.Now, choose a countable dense subset {(x 1 , v 1 ), (x 2 , v 2 ), . ..} in Ω and define where N * is the adjoint operator of N .The or- can be derived from the process of Gram-Schmidt orthogonalization of {Ψ i } ∞ i=1 as , and Proof.We acquire i=1 is dense in Ω.Therefore, we obtain N f = 0. From the existence of N −1 , it follows that f = 0.The proof is completed.
i=1 is dense in Ω, then the solution of the problem is obtained as: . Therefore, we get This completes the proof.Now the approximate solution f n can be obtained from the n-term intercept of the exact solution f and Obviously then, we have 2 is monotonically decreasing in n.
Proof.We have Therefore, we obtain Furthermore, we have To test the accuracy of the reproducing kernel Hilbert space mehod, an example has been given.The results are compared with the exact solutions.Let us take into consideration the problem of obtaining (f (x, v), σ(x, v)) in Ω = (−1, 1) × (1, 2).The exact solution of the problem is given as [1]: Using our technique, we choose 25, 64 and 100 points in the region Ω = [−1, 1] × [1, 2] and obtain f 25 , f 64 and f 100 .Numerical results are in good agreement with the exact solution.In order to prove the convergence of the exact solution we found absolute errors for different values of dense points n.We give the maximum absolute errors for different number of dense points in Table 1.The results demonstrate that the errors become smaller as n increases.

Conclusion
In this work, the reproducing kernel Hilbert space method was implemented for solving an inverse problem for the kinetic equation.Given technique is demonstrated to be of good convergence.It seems that this technique can also be applied to higher dimensional inverse problems.We found the reproducing kernel functions for solutions of a coefficient inverse problem for the kinetic equation.We concluded that these reproducing kernel functions can be used in much more complicated problems.We demonstrated our results by a table.These results proved the power of the reproducing kernel Hilbert space method.