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SIAMSEAS2015

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SIAMSEAS2015

  1. 1. The application of Homotopy Analysis Method for the solution of time-fractional diffusion equation with a moving boundary Ogugua N. Onyejekwe Department of Mathematics Indian River State College 39th Annual SIAM Southeastern Atlantic Section Conference March 20-22 2015 Ogugua N. Onyejekwe Homotopy Analysis Method
  2. 2. Abstract It is difficult to obtain exact solutions to most moving boundary problems. In this presentation we employ the use of Homotopy Analysis Method(HAM) to solve a time-fractional diffusion equation with a moving boundary condition. HAM is a semi-analytic technique used to solve ordinary, partial, algebraic, fractional and delay differential equations. This method employs the concept of homotopy from topology to generate a convergent series solution for nonlinear systems. The homotopy Maclaurin series is utilized to deal with nonlinearities in the system. Ogugua N. Onyejekwe Homotopy Analysis Method
  3. 3. Abstract HAM was first developed by Dr. Shijun Liao in 1992 for his PhD dissertation in Jiatong University in Shangia. He further modified this method in 1997 by introducing a convergent - control parameter h which guarantees convergence for both linear and nonlinear differential equations. Ogugua N. Onyejekwe Homotopy Analysis Method
  4. 4. Abstract There are advantages to using HAM [4] it is independent of any small/large physical parameters. when parameters are chosen well, the results obtained show high accuracy because of the convergence- control parameter h. there is computational efficiency and a strong rate of convergence. flexibility in the choice of base function and initial/best guess of solution. Ogugua N. Onyejekwe Homotopy Analysis Method
  5. 5. Parameters s (t) - diffusion front C0 - initial concentration of drug distributed in matrix. Cs - solubility of drug in the matrix C (x, t) - concentration of drug in the matrix ℘ - diffusivity of drug in the matrix (assumed to be constant) Dα t - Caputo Derivative R - scale of the polymer matrix Ogugua N. Onyejekwe Homotopy Analysis Method
  6. 6. Problem Definition Figure 1: Profile of concentration. The first picture is the initial drug loading. The second picture is the profile of concentration of the drug in the matrix at time t.[10] Ogugua N. Onyejekwe Homotopy Analysis Method
  7. 7. Assumptions We will only consider the early stages of loss before the diffusion front moves closer to R and assume that C0 > Cs. Perfect sink is assumed. Ogugua N. Onyejekwe Homotopy Analysis Method
  8. 8. Introduction Given the domain WT = {(ξ, t) : 0 < ξ < s (t) , 0 < α ≤ 1, t > 0} (1) The following problem is considered Dα t C (ξ, t) = ℘ ∂2C (ξ, t) ∂ξ2 , (2) with the initial condition C (ξ, 0) = 0 (3) and the following boundary conditions C (s (1) , 1) = k1, C (s (t) , t) = Cs, t > 0, (4) where k1 is any constant. Ogugua N. Onyejekwe Homotopy Analysis Method
  9. 9. (C0 − Cs) Dα t s (t) = ℘ ∂C (ξ, t) ∂ξ (ξ = s (t) , t > 0) , (5) s (1) = k2 (6) k2 is a constant that depends of the value of α where Dα t is defined as the Caputo derivative Dα t f (t) = t 0 (t − τ)n−α−1 Γ (n − α) fn (τ) dτ, (α > 0) , (7) for n − 1 < α < n, n ∈ N and Γ ( ) represents the Gamma function. Ogugua N. Onyejekwe Homotopy Analysis Method
  10. 10. Reducing Governing Equations to Dimensionless Variables The reduced dimensionless variables are defined as x = ξ R , τ = ℘ R2 1 α t, u = C Cs , S (τ) = s (t) R (8) Ogugua N. Onyejekwe Homotopy Analysis Method
  11. 11. Reducing Governing Equations to Dimensionless Variables The governing equation (2) subjected to conditions (3) − (5) can be reduced to the dimensionless forms Dα t u (x, τ) = ∂2u (x, τ) ∂x2 (0 < x < S (τ) , τ > 0) (9) u (S (1) , 1) = 1 (10) where S (1) varies for different values of α and η u (x, τ) = 1, (x = S (τ)) , τ > 0, (11) ∂u (x, τ) ∂x = ηDα t S (τ) , (x = S (τ)) , τ > 0, (12) where η = C0−Cs Cs Ogugua N. Onyejekwe Homotopy Analysis Method
  12. 12. Solution by HAM To solve equation (9) by homotopy analysis method, the the initial guess for u (x, τ) is chosen as u0 (x, τ) = (a0)−1 xτγ1 (13) where a0 = Γ(1−α 2 ) ηΓ(1+α 2 ) 1 2 , γ1 = −α 2 The initial guess for the diffusion front is chosen as S0 = a0τ α 2 (14) Ogugua N. Onyejekwe Homotopy Analysis Method
  13. 13. Solution by HAM The auxiliary linear operator is L [φ (x, τ; q)] = ∂2φ (x, τ) ∂x2 (15) with the property L [k] = 0 (16) where k is the integral constant, φ (x, τ; q) is an unknown function. The nonlinear operator is given as N [φ (x, τ; q)] = ∂2φ (x, τ; q) ∂x2 − ∂αφ (x, τ; q) ∂τα (17) Ogugua N. Onyejekwe Homotopy Analysis Method
  14. 14. Solution by HAM By means of HAM,defined by Liao, we construct a zeroth-order deformation (1 − q) L [φ (x, τ; q) − u0 (x, τ)] = qhN [φ (x, τ; q)] , (18) where q ∈ [0, 1] is the embedding parameter, h = 0 is the convergence-control parameter,u0 (x, τ; q) is the initial/best guess of u0 (x, τ) Ogugua N. Onyejekwe Homotopy Analysis Method
  15. 15. Solution by HAM Expanding φ (x, τ; q) in Taylor series with respect to q we obtain, φ (x, τ; q) = u0 (x, τ) + +∞ m=1 um(x, τ)qm (19) Clearly we see that when q = 0 and q = 1 equation (19) becomes φ (x, τ; 0) = u0 (x, τ) , φ (x, τ; 1) = u (x, τ) (20) If the auxiliary linear operator L, the initial guess u0 (x, τ) and the convergence-control parameter h are properly chosen so that the series described in (20) converges at q = 1, then u (x, τ) will be one of the solutions of the problem we have considered. Ogugua N. Onyejekwe Homotopy Analysis Method
  16. 16. Solution by HAM Differentiating the zero-order deformation equation (18) m times with respect to q and then dividing it by m! and finally setting q = 0 , we obtain an mth-order deformation equation L [um (x, τ) − χmum−1 (x, τ)] = hVm −−−−−−−→ um−1 (x, τ) (21) where χm = 0 if m ≤ 1; 1 if m > 1. and Vm −−−−−−−→ um−1 (x, τ) = ∂2um−1 (x, τ) ∂x2 − ∂αum−1 (x, τ) ∂τα (22) Ogugua N. Onyejekwe Homotopy Analysis Method
  17. 17. Solution by HAM We have um (x, τ) = χmum−1 (x, τ) + hL−1 Vm −−−−−−−→ um−1 (x, τ) + k (23) and the integration constant k is determined by the boundary condition equation (10). Looking at equation (23), the values for um (x, τ) for m = 1, 2, 3, ... can be obtained and the series solution gained. Ogugua N. Onyejekwe Homotopy Analysis Method
  18. 18. Solution by HAM The approximate analytic solution is gained by truncating the following series u (x, τ) = m i=0 ui(x, τ). (24) Equation (24) contains the convergence-control parameter h, which determines the convergence region and rate of the homotopy-series solution. The convergence-control parameter h is obtained by setting (u(S (1) , 1)HAM = (u(S (1) , 1)exact The diffusion front S (τ) is obtained by setting (u(S (τ) , τ))HAM = 1 Ogugua N. Onyejekwe Homotopy Analysis Method
  19. 19. Comparison between Approximate and Exact Solutions when x=0.75 The exact solution for u (x, τ) and S (τ) are given as follows [10]. u (x, τ) = H ∞ n=0 x τ α 2 2n+1 (2n + 1)!Γ 1 − 2n+1 2 α (25) S (τ) = p.τ α 2 (26) Ogugua N. Onyejekwe Homotopy Analysis Method
  20. 20. Comparison between Approximate and Exact Solutions when x=0.75 Figure 2: Drug Distribution in tissue when η = 0.5 and α = 1, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  21. 21. Comparison between Approximate and Exact Solutions when x=0.75 Figure 3: Diffusion Front in tissue when η = 0.5 and α = 1, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  22. 22. Comparison between Approximate and Exact Solutions when x=0.75 Figure 4: Drug Distribution in tissue when η = 0.5 and α = 0.75, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  23. 23. Comparison between Approximate and Exact Solutions when x=0.75 Figure 5: Diffusion Front in tissue when η = 0.5 and α = 0.75, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  24. 24. Comparison between Approximate and Exact Solutions when x=0.75 Figure 6: Drug Distribution in tissue when η = 0.5 and α = 0.5, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  25. 25. Comparison between Approximate and Exact Solutions when x=0.75 Figure 7: Diffusion Front in tissue when η = 0.5 and α = 0.5, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  26. 26. Comparison between Approximate and Exact Solutions when x=0.75 Figure 8: Drug Distribution in tissue when η = 1 and α = 0.5, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  27. 27. Comparison between Approximate and Exact Solutions when x=0.75 Figure 9: Diffusion Front in tissue when η = 1 and α = 0.5, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  28. 28. Comparison between Approximate and Exact Solutions when x=0.75 Figure 10: Drug Distribution in tissue when η = 3 and α = 0.5, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  29. 29. Comparison between Approximate and Exact Solutions when x=0.75 Figure 11: Diffusion Front in tissue when η = 3 and α = 0.5, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  30. 30. Comparison between Approximate and Exact Solutions when x=0.75 Figure 12: Drug Distribution in tissue when η = 9 and α = 0.5, u (x, τ)HAM is in red and u (x, τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  31. 31. Comparison between Approximate and Exact Solutions when x=0.75 Figure 13: Diffusion Front in tissue when η = 9 and α = 0.5, S (τ)HAM is in red and S (τ)EXACT is in green Ogugua N. Onyejekwe Homotopy Analysis Method
  32. 32. Conclusion When calculating the values for u (x, τ) for a fixed value of η and varying values of α, the higher the value of α, the smaller the relative error. For fixed values of η and varying values of α, the approximate and exact values of S (τ) are in direct agreement with each other. Similarly for fixed values for α and varying values of η, the approximate and exact values of S (τ) are in direct agreement with each other. Whereas for fixed values for α and varying values of η, the values of u (x, τ) are in more agreement than they were for u (x, τ) for a fixed value of η and varying values of α. The relative error is smaller. Ogugua N. Onyejekwe Homotopy Analysis Method
  33. 33. Conclusion We have shown that HAM can be used to accurately predict drug distribution in tissue u (x, τ) and the diffusion front S (τ) for different values of α and η. Ogugua N. Onyejekwe Homotopy Analysis Method
  34. 34. References I A.K. Alomari. Modifications of Homotopy Analysis Method for Differential Equations: Modifications of Homotopy Analysis Method, Ordinary, Fractional,Delay, and Algebraic Equations. Lambert Academic Publishing,Germany, 2012. S. Liao. Homotopy Analysis Method in Nonlinear Equations. Springer,New York, 2012. S. Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC,New York, 2004. Ogugua N. Onyejekwe Homotopy Analysis Method
  35. 35. References II S.Liao Advances in The Homotopy Analysis Method World Scientific Publishing Co.Pte. Ltd, 2014. Rajeev, M.S. Kushawa Homotopy perturbation method for a limit case Stefan Problem governed by fractional diffusion equation. Applied Mathematical Modeling,37(2013),3589-3599. S.Das, Rajeev Solution of Fractional Diffusion Equation with a moving boundary condition by Variational Iteration and Adomain Decomposition Method. Z. Naturforsch,65a(2010), 793-799. Ogugua N. Onyejekwe Homotopy Analysis Method
  36. 36. References III S. Liao. Notes on the homotopy analysis method - Some definitions and theorems. Common Nonlinear Sci.Numer.Simulat, 14(2009),983-997. V.R.Voller An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. International Journal of Heat and Mass Transfer,53(2010),5622-5625. V.R.Voller, F.Falcini, R.Garra Fractional Stefan Problems exhibiting lumped and distributed latent-heat memory effect. Physical Review,87(2013),042401. Ogugua N. Onyejekwe Homotopy Analysis Method
  37. 37. References IV X.Li, M.Xu, X.Jiang. Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition. Applied Mathematics and Computation,208(2009),434-439. X.Li, S.Wang and M.Zhao Two methods to solve a fractional single phase moving boundary problem. Cent.Eur.J.Phys.,11(2013),1387-1391. Ogugua N. Onyejekwe Homotopy Analysis Method

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