Determination of an Unknown Diffusion Coefficient in a Parabolic Inverse Problem

................................................................................... iii ÖZ.................................................................................................. iv DEDICATION.................................................................................. v ACKNOWLEDGEMENT.................................................................... vi LIST OF TABLES.............................................................................. ix LIST OF FIGURES............................................................................ x

Anahtar kelimeler: sonlu fark yöntemleri, parabolik ters problemi sorun, yakınsama, hata tahminleri, maksimum ilkesi. v To My Family vi ACKNOWLEDGEMENT I give thank and praise to Almighty Allah for his marcy, guidance and blessing upon me for making this thesis work to reality.My deepest gratitude is to my supervior Ass.prof.Dervis Subasi I have been amazingly privileged to have an advisor who give me the sovereignty to studay on my own and at the same time the assist to recuperate when my step weakened.His tolerance and support help me overwhelmed various crises condition and finish this thesis.I hope that one day I would become as good an advisor to my student as Subasi has been to me.
Most essentially, none of this would have been likely possible without the prayer and support of my family.My family to whome this thesis is dedicate to, has been a constant source of love, concern and support all these period.I would like to express my heart-felt gratitude to my family, I have to give a special mention, for the support given by: my parent and my brother Engr.Sani G Ibrahim I really I appreciate the generosity, love, guidance and counseling may almighty Allah reward them abundently.
Finally I wish to thanks all my friends who helped and encouraged me during the period of my studies. vii

Chapter 1 INTRODUCTION
In this thesis we analyse the problem of solving two unknown functions ( ) and the diffusion coefficient ( ) in the parabolic inverse problem where achieved from the continuity techniques.In ,the numerical solution of ( ) ( ) are also debated using Explicit, Implicit and Crank Nicolson numerical schemes and higher order was recommended to determine the function and the unknown time reliant on coefficient ( ), in which so many numerical investigation were obtained to examine the effectiveness and accuracy of the numerical consequence, error approximation and numerical solution of ( ) and ( ) were developed.In , -Pseudospectral Legendre scheme is engaged to solve problem ( ) ( ) where the Errors of ( ) and ( ) are acquired by using Explicit, implicit, Crank Nicolson, Saulyev's first and second kind.In [4] the author discussed over the problem of determining concurrent time reliant on thermal diffusivity and the temperature circulation in one dimensional parabolic equation in nonlocal boundary and integral over resolve conditions, the uniqueness and existence condition of classical clarification of the problem were also discussed.In ,finite difference estimate to an inverse problems ( ) ( ) were also deliberated, the Implicit Euler scheme is considered and is shown that the scheme is stable using maximum norm and convergence are proved using discrete maximum principle.The error estimation and numerical investigation of ( ) and ( ), and some newly projected procedure are presented.Author in ,also researchED on the problem

Transformation of the Inverse Problem
Taking the derivative of equation ( ) with the respect to , we obtained Substituting equation ( ) in eqution ( ) we have Now equations ( ) ( ) change to the resulting problem Our process is based on the following alteration, by setting taking derivative of ( ) with respect to we have Therefore For initial condition at , Second derivative of ( ) with the respect to , yield to (2.10) Transformation of the left and right boundary condition at and , respectively, we know for the left boundary condition It implies that Similarly, for the right boundary condition Then it implies that

Backward Time Centered Space (BTCS)
Backward time centered space can be defined using the forward derivative approximation for the time derivative and second order approximation for the spatial derivative defined at the point ( ).Then the overall approximation is called Backward Time Centered space or Backward Euler scheme.

Lemma 2.1 [6]
Suppose that ( ) ,and there exist such that where Now the backward time centered space (see Figure 1) can now be defined by ( Proof: The inequality ( ) follows from Taylor's expansion, also the inequality ( ) hold from smoothness of the data and .Finally, the inequalities ( ) follows from ( ), corollary and ̅ .

Let ( )
where ( ).Then from ( ) we have Proof: the inequality follows from ( ) and the definition of , Upon using the transformation of ( ) and ( ) with some simple calculation, we find out that satisfies Step 3: Let and , , ( ) or for sufficiently large and for some ( ) and suffiently small such that and such that for Therefore we can clearly see that It implies that ( ) where depends on and .By the same argument we can deal with and we can obtain the equivalent inequality.
Step 5: It's enough to see from ( ) and ( ) that where depends on and .Finally from equation ( ) and ( ) which completes the proof.

Chapter 4 NUMERICAL RESULTS AND DISCUSSION
In this chapter, we will present the numerical experiment from solving two model problems by using the numerical procedures discussed in the previous chapters in    Chapter 5

CONCLUSION AND FUTURE WORK
In this thesis Backward time centered space of the finite difference scheme were applied for recovering time dependent diffusion coeffient in one-dimensional parabolic inverse problem.The suggested numerical approaches for solving these two model problems are very reasonable and these test experiment backed our theoretical expectation.Using the backward time centered space formula for the one -dimensional diffusion problem with an additional measurement define our model well.Various of issues can be tendent as subject for future examinations in this field.
We can mentioned some of them in the following: We can extend this research to two or three dimensional problems, Employing Crank Nicolson finite difference techniques to solve the current problems, we can also extend to higher-order accurate finite difference methods, we can also apply on explicit formula which is conditionally stable, dealing with the more difficult extra measuments, using new numerical measures for solving Backward time centered space problems by using the descrived methods for simplifying the present problem with the Neumann's boundary condition.

(
) ( ) , but the numerical results of the investigation are far-off from tolerable.In[7] Cannon and Jones studied ( ) subject to time reliant on boundary conditions.The foremost target of the research is to decrease the problematic case to nonlinear integral equation for the quantity ( ).This suggestion, which depends on the explicit arrangement of elementary solution of the heat equation, does not simply lead to the separation of m space variable forIn[8] Cannon and William verified the fortitude of a time reliant on conductivity for potential arbitrary field in , their technique can be labelled as a "lenient" amendment of the methodology of Jones, and depends on the compactness and maximum principle of a convinced smoothing to produce a desire effect by sequential estimates.This thesis is prepared as follows.In Chapter 2, the finite difference method is expressed from the renovation of parabolic inverse problem and several elementary basis are indicated in the form of lemmas.The backward time centered space (BTCS) is considered and it is shown to be stable in the maximum norm by means of discrete form of the maximum principle for parabolic finite difference scheme .In Chapter 3, the convergence and error estimate of the numerical method of the transformed parabolic inverse problem is discoursed.In chapter 4, two numerical investigations accessede to determine or to check the correctness and efficiency of the backward Euler estimates by presenting the errors for ( ) ( ) of each models.Finally conclusion and observation are given for each experiment.In this chapter the numerical methods of one dimensional parabolic inverse problem is advanced to solve (1.1)-(1.4)The finite difference consequent from substituting the space and the time derivative.The parabolic space domain , -, is derived in to mesh of with the spatial step size and the time step size .Now we can design the grid points ( ) by where and are any integers, the notation are used to designate the finite difference estimates of ( ) ( )

Figure 1 .
Figure 1.Computational molecule for BTCS order to give clear overview of the approaches.Each model problem we used various values of with the fixed and the point choose as an interior point of the domain for the two model problems.In order to verify the accuracy of ( ) and ( ) using proposed finite difference schemes the following error calculation are used

Figure 6 :
Figure 6: Absolute error for ( ) and at each time step

Table 8 :
Approximate result of ( ) with the various ……………..…………… ix This chapter is apprehensive with the conditions that must be gratified if the solution of the finite difference equation is to be sensibly correct estimate to the solution of the correspondent parabolic partial differential equation.These conditions are attendant with the two different but interrelated problems.The first concerns the convergence of the exact solution of the approximating difference equations to the solution of the differential equation, the second concerns the unbounded growth or measured decay of any errors associated with the solution of the finite difference equations.Therefore convergence estimate theorem can be stated below.

Table 1 :
Exact and Approximate values of with , and

Table 3 :
Exact and Approximat values for ( ) with the

Table 4 :
Exact and Approximate values for ( ) with at

Table 6 :
Exact and approximate values of with and

Table 7 :
Exact and approximate values of ( ) for at Figure 10: Absolute errors for ( ) for at each time level Figure 11: Absolute errors of ( ) for at each time step