SINE TRANSFORMATION FOR REACTION DIFFUSION CONTROL PROBLEM

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA IAŞI Tomul LII, s.i, Matematică, 6, f. SINE TRANSFORMATION FOR REACTION DIFFUSION CONTROL PROBLEM BY J.O. OMOLEHIN Abstract. The Reaction Diffusion Equations system is transformed into an optimization problem and the result control problem has opened us to the possibility that will allow us to study Reaction Diffusion Equations and their numerically applications. Specifically, the modified version of Conjugate Gradient Method (CGM) algorithm can be used as a method of solving for some of its applications. Mathematics Subject Classification : 49A. Key words: Diffusion, conjugate, gradient, reaction, control.. Introduction. This is an first attempt to bring the concept of optimization theory and application into Reaction Diffusion Equations. The work concerns with the transformation of linear reaction diffusion equations into a control problem so that optimization techniques can be used for solving the resulting problem. This new area of research is designated by the control of reaction diffusion equations. Basically, this work considers the transformation of reaction diffusion system into the quadratic cost functional with differential equations as constraints. This type of problem can be solved through penalty function method, specifically, the E.C.G.M. algorithm due to Ibiejugba[7] can be used as a method of solution. Once the existence and uniqueness of solution is established a control operator based on the formalism of the construction of operator A in the conventional Conjugate Gradient Method (CGM) algorithm [6] can be constructed explicitly. Our result is stated in section.

2 6 J.O. OMOLEHIN For the reaction diffusion equations considered in [5] there are no explicit or analytic form of solution. However, Hill [5] outlined a general procedure for obtaining a closed form representations of the solutions u(x, t) and v(x, t) for the linear reaction diffusion equations: (.) u t = D u au + bv v t = D v + cu dv where D, D, a, b, c and d are all non-negative constants. Hill [5] showed that closed form solutions of (.) can be given in term of integral arbitrary heat functions h (x, t) and h (x, t). These functions satisfy the classical heat equation (.) h t = h, In particular he established that the formal solutions of (.) are u(x, t) = e at h (x, D t) (.3) + b e λt (D D ) D t D t e {e µξ (ξ D t) D t ξ I(n)h (x, ξ) +b I (n)h (x, ξ) dξ (.4) v(x, t) = e at h (x, D t) + c e λt (D D ) D t D t e {e µξ (D t ξ ) ξ D t I(τ)h (x, t) +c I (τ)h (x, ξ) dξ where the constants λ and µ are given by (.5) λ = (ad dd ) D D (.6) µ = (a d) D D

3 3 SINE TRANSFORMATION 7 I o, I are the usual modified Bessel functions and τ is given by (.7) τ = (bc) (D D ) [(D t ξ)(ξ D t)]. Hill [5] considered the application of his general formulae to the stability problem arising from a model of an arms race which incorporates the features of deteriorating armaments. The situation is as follows: In [5], Richardson proposed that the military spending of two nations locked in an arms race can be modelled by the following linear system (.8) (.9) dp (t) = ap(t) + bq(t) + g dt dq (t) = cp(t) dq(t) + h dt where p(t) and q(t) denote armament levels of the two nations at time t and a, b, c, d, g and h denote positive constants. The constants b and c are called Threat coefficients and they signify the degree to which a nation is stimulated by another nation s weapon stock to increase her own stocks. The constants a and d are called fatigue coefficients and are a measure of the prevailing economic circumstances which inhibit armament buildup. The constants g and h denote a measure of the circumstances which prevents a complete disarmament in the situation when both nations have zero armaments. A balance of power situation results when the armament level remains constant over a long period of time and these levels are given in [5] by the following equations (.) p = (.) q = gd + hb (ad bc) gc + ha (ad bc) (ad bc) > In [5], Gopalsamy developed Richardson model and proposed that the armament levels p(x, t) and q(x, t) satisfy (.) p t + e p x = δ δ p ap + bq + g x

4 8 J.O. OMOLEHIN 4 and (.3) q t + e q x = δ q cq + dp + h x where e, e, δ and δ denote positive constants and the remaining constants are as previously defined. Hill [5] further developed the model and asserted that in order to investigate the stability of power situation (p, q ) given by (.) and (.) he sets (.4) p(x, t) = p + u(x, t) and (.5) q(x, t) = q + v(x, t) so that from (.) and (.3) u t = D u x e u au + bv, x t v t = D v x e v cu + dv, x t (.6) D i = δ i (i =, ) u(x, ) =, v(x, ) = u(, t) = u, v(, t) = v u(x, t), v(x, t) as x. We shall now transform the whole of reaction diffusion system (.6) into a control problem in our main result.. Main result Theorem.. The reaction diffusion system (.6) can be transformed into a quadratic cost functional of the form: Minimize t {v (t)+v (t)+...+v n(t)+u (t)+u (t)+... u n(t)dt subject to u i (t) v i (t) = Cv i (t) + Du i (t), where C = D π i + d + b, D = D π i a c, i =,..., n. Proof. We proceed as follows. Consider the problem:

5 5 SINE TRANSFORMATION 9 Minimize [u (x, t) + v (x, t)]dxdt subject to: (.7) u t = D u x e u au + bv = x v t = D v x e v cu + dv =, t x with the following initial and boundary conditions: u(x, ) =, v(x, ) = u(, t) = u, v(, t) = v u(x, t), v(x, t) as x u(x, ) t = v(x, ) t where the boundary condition at x equals zero represents the fact that both nations looked in the arms race are maintaining a constant level of perfect undeteriorated strategic weapon system and the integral given by (.7) is a measure of the cost. To obtain explicit solutions of these boundary value problems (.6) Gopalsamy in ref. [5] assumed that e = e, D = D and a = d. We also adopt in this study these values for simplicity and consistency. Let v(x, t) = u(x, t) = v i (t) = sin πix, x sin πix u i (t), x t. Thus, we have v t (x, t) = u t (x, t) = v i (t) sin πix u i (t) sin πix

6 J.O. OMOLEHIN 6 v xx (x, t) = π i v i (t) sin πix u xx (x, t) = π i u i (t) sin πix v (x, t) = u (x, t) = vi (t) sin πix u i (t) sin πix Substituting the values of v (x, t), u (x, t)in the integral of (.7). We obtain = = = = = = [v (x, t) + u (x, t)]dxdt [ vi (t) sin πix + [ vi (t)[ cos πix] + ] u i (t) sin πix dxdt ] u i (t)[ cos πix] dxdt { [ ] sin πix [ ] sin πix vi (t) + u i (t) dt πi πi { vi (t)[ (+ )] + u i [ (+ )](t) dt { vi (t) u i dt { v (t) + v (t) v n + u (t) + u (t) u n dt. Since it is a minimization problem the can be omitted. Thus, we have factor in the right hand side [v (x, t) + u (x, t)]dxdt

7 7 SINE TRANSFORMATION = {v(t) + v(t) vn(t) + u (t) + u (t) u n(t)dt. Following the idea of Gopalsamy in ref.[5] we set e = e =. Equating the constraints in (.7) we obtain δu δt D δ u δx + δu δv + au bv = δx δt D δ v δx + δv cu + dv. δx Next substituting values for u t, v t, u xx, v xx, u, v in the last equation, we obtain u i (t) sin πix + D πi sin πix + a u i sin πix b v i (t) sin πix = v i (t) sin πix + D π i sin πix c u i sin πix + d v i (t) sin πix Therefore, by dividing both sides by sin πix we obtain u i (t) + D π i + a u i b v i (t) = c u i (t) + d v i (t) v i (t) + D i since sin πix, < x <. Rearranging and dropping the summation sign, we obtain u (t) v (t) = [D π + d + b]v (t) + [ D π a c]u (t) u (t) v (t) = [D π + d + b]v (t) + [ D π a c]u (t) u n (t) v n (t) = [D π n + d + b]v n (t) + [ D π n a c]u n (t). We can put it in a compact form in the following manner: u i (t) v i (t) = Cv i (t)+du i (t), where C = D π i +d+b, i =,..., n, D = D π i a c, i =,..., n.

8 J.O. OMOLEHIN 8 Consequently, problem (.7) reduces to Minimize t {v (t)+v (t)+...+v n(t)+u (t)+u (t)+... u n(t)dt subject to u i (t) v i (t) = Cv i (t) + Du i (t), where C = D π i + d + b, i =,..., n, D = D π i a c, i =,..., n. Hence we have our result. REFERENCES. Aifantis, E.C. A new interpretation of Diffusion in High Diffusivity paths- A continum Approach, Acta Metalurgica, 7(979), Aifantis, E.C. Continuum basis for diffusion in regions with multiple diffusivity, J. Appl. Phys. 5(3), March, Hestenes, M.; Stiefel, E. Method of conjugate Gradients for solving linear systems, J. Res. Nat. Bus. standards, 49(95), Hill, J.M. An integral Equation Arising in Double Diffusion Theory, J. Maths. Mathematical Analysis and Applications 77(98), Hill, J.M. On the solution of Reaction Diffusion Equations, IMA Journal of Applied Mathematics, 7(977), Ibiejugba, M.A. Computing Methods is Optimal Control, University of Leeds, Leeds, England, PhD. Thesis, Ibiejugba, M.A.; Onumanyi, P. On control operators and some of its applications, Journal of Mathematical Analysis and Applications, 3(984), Liusternik, L.A.; Sobolev, V.I. Element of Functional Analysis, Frederic Ungar, English Translation, N.Y., Omolehin, J.O. Numerical Experiments with Extended Conjugate Gradient Method Algorithm, University of Ilorin, Nigeria, MSc. Thesis, Omolehin, J.O. On the Control of Reaction Diffusion Equations, Univesity of Ilorin, Nigeria, PhD, Thesis, 99. Received: 6.I.5 Department of Mathematics, University of Ilorin, Ilorin, NIGERIA, omolehin joseph@yahoo.com