In this article Homotopy Perturbation Method (HPM) is applied to obtain an approximate analytical solution of a fractional diffusion equation with an external force and a reaction term different from the reaction term used by Das and Gupta (2010). The anomalous behavior of diffusivity in presence or absence of linear external force due to the presence of this force of reaction term are obtained and presented graphically

Das, S.

Gupta, P. K.

Kumar, R.

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 5 Issue 3 Special Issue 1 – August 2010 Article 9 8-2010 An Approximate Analytical Solution of the Fractional Diffusion Equation with External Force and Different Type of Absorbent Term - Revisited S. Das Banaras Hindu University R. Kumar Banaras Hindu University P. K. Gupta Banaras Hindu University Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Analysis Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons, and the Probability Commons Recommended Citation Das, S.; Kumar, R.; and Gupta, P. K. (2010). An Approximate Analytical Solution of the Fractional Diffusion Equation with External Force and Different Type of Absorbent Term - Revisited, Applications and Applied Mathematics: An International Journal (AAM), Vol. 5, Iss. 3, Article 9. Available at: https://digitalcommons.pvamu.edu/aam/vol5/iss3/9 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. 101  Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466  Special Issue No. 1 (August 2010)  pp. 101 – 109  Applications and Applied Mathematics:  An International Journal (AAM)     An Approximate Analytical Solution of the Fractional Diffusion Equation with External Force and Different Type of Absorbent Term – Revisited   S. Das, R. Kumar and P. K. Gupta Department of Applied Mathematics Institute of Technology Banaras Hindu University Varanasi – 221 005   Received: May 15, 2010; Accepted: July 5, 2010   Abstract In this article Homotopy Perturbation Method (HPM) is applied to obtain an approximate analytical solution of a fractional diffusion equation with an external force and a reaction term different from the reaction term used by Das and Gupta (2010). The anomalous behavior of diffusivity in presence or absence of linear external force due to the presence of this force of reaction term are obtained and presented graphically.   Keywords: Fractional diffusion equation; External force; Reaction term; Absorbent term; Homotopy Perturbation Method.                MSC (2010) No: 26A33; 34A25; 35R11; 60G22.   1. Introduction We focus our attention to find the solution of the equation  )1(,10,),()()],()([),(),(022dxxuttxuxFxtxuxDtxutt  1Das et al.: Approx Analytical Solution of the Fractional Diffusion EquationPublished by Digital Commons @PVAMU, 2010102  Das et al. where, D  is a diffusion coefficient, )(xF  is an external force, )(t  is a time-dependent absorbent term which may be related to a reaction diffusion process. Here the reaction term dxxutt),()(0  is different from that used by Schot et al. (2007) and Das and Gupta (2010). Here we have used the fractional Riemann-Liouville fractional integral operator of order 10 , 0t .  Also, the Caputo fractional derivative, applied to the time variable is defined by  tmmtdUmtUD01)()()()(1)( ,   mm 1 , Nm , .0t    In this article the Homotopy Perturbation Method He (1999, 2000) is used to solve the fractional diffusion equation problem in the presence of both linear external force and an absorbent term. Using the initial condition, the approximate analytical expressions of ),( txu for different Brownian motions are obtained. The objective of the study is to show the effect of reaction term on the fractional diffusion equation with or without the presence of the linear external force.  2. Solution of the problem Our aim is to solve the analytical fractional diffusion equation (1) for 1D and .)( xkxF  The equation (1) now becomes    dxxuttxuxxkxtxuttxut),(.)()),((),(),(022 (2) with initial condition    ).()0,( xfxu                      (3)  Equation (2) can be written in operator form as  txxxt dxxuttxuktxuDxktxuDtxuD0),(.)(),()),((),(),( (4) where, .tDt   According to the homotopy perturbation method, we construct the following homotopy 2Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 9https://digitalcommons.pvamu.edu/aam/vol5/iss3/9AAM: Intern. J., Special Issue No. 1 (August 2010)  103   ],),(.)(),()),((),([),(0txxxt dxxuttxuktxuxDktxuDptxuD (5) where the homotopy parameter p  is considered as a small parameter ( ]1,0[p ). In case 0p , equation (5) becomes a linear equation, 0uDt , which is easy to solve Belendez et al. (2008), Darvishi and Khani (2008), Mausa and Ragab (2008). Now applying the classical perturbation technique, we can assume that the solution of equation (2) can be expressed as a power series in p as given below:   .),(),(),(),(),(),( 44332210 txuptxuptxuptxuptxutxu  (6) When 1p , equation (5) corresponds to equation (4) and (6) becomes the approximate solution of (4) i.e., of equation (2). The convergence of the method has been proved in He (2000). Substituting equation (6) for equation (5), and equating the terms with the identical powers of p, we can obtain a series of equations:   0p : 0t)(x,0uDt   (7)  1p : txxxt dxxutuuDxkuDuD000001),(.)(t)(x,kt))(x,(t)(x,t)(x, (8)  2p : txxxt dxxutuuDxkuDuD011112),(.)(t)(x,kt))(x,(t)(x,t)(x, (9)  3p : txxxt dxxutuuDxkuDuD022223),(.)(t)(x,kt))(x,(t)(x,t)(x, (10) and so on.  Applying the operator tJ (the inverse of Caputo operator tD ) on both sides of the equations (7) – (10), we obtain the solutions of 0i,t)(x,iu for )()(1tt  , 0 . 10  (Schot et al. (2007)) as   )(),(0 xftxu   (11)  )12()(')1())()(')(''(),(21txftxfkxfxkxftxu (12) 3Das et al.: Approx Analytical Solution of the Fractional Diffusion EquationPublished by Digital Commons @PVAMU, 2010104  Das et al.  )12())()('3)()4()(2)((),(222)2(22)3()4(2txfkxfxkxfkxkxfxkxftxu                 )14()('')13())('3)(2)(2(423)2()3( txftxfkxfxkxf (13)  )()15()()93()(3)((),( )3(233)4(22)5()6(3 xfxkxkxfkxkxfxkxftxu                   )(6)(3()13())()('7)()136( )4()5(333)2(223 xfxkxftxfkxfxkxfkxk                   )((3)14())('7)(12)()153( )4(242)2(2)3(22 xftxfkxfxkxfkxk                   )16()()15())(2)(6)3(35)2()3( txftxfkxfxk  (14) where,      ))(()()( xfxxfrrr ,   1r . Proceeding in this manner the rest of the components of 0,nun  can be completely obtained, and the series solutions are thus entirely determined.  Finally, we approximate the analytical solution of ),( txu by the truncated series    ),,(lim),( txtxu NN   (15) where  .),(),(10NnnN txutx The above series solutions generally converge very rapidly. A classical approach of convergence of this type of series is already presented by Abbaoui and Cherruault (1995).     4Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 9https://digitalcommons.pvamu.edu/aam/vol5/iss3/9AAM: Intern. J., Special Issue No. 1 (August 2010)  105  3. Particular Cases  Case I. If 1,0,)( kxxf i.e., in the presence of only external force, the expression of the displacement becomes   )13(8)12(4)1(21),(32 tttxtxu              0 )1(2rrrrtx                         ),2( tEx  (16) where 0 )1()(rrrttE  , ( 0 ) is the Mittag-Leffler function in one parameter. This result is same as the results given in Das and Gupta (2010), Saha Ray and Bera (2006) and Das (2009).   Case II.  If 1,1,)( kxxf  i.e., in the presence of both the linear external force and absorbent term,   )14(7)13(3)12()13(8)12(4)1(21),(3232 ttttttxtxu           0)1(0 )1)1(()12()1(2rrrrrrrttrtx           ),()2( 1, tKEttEx   (17) where, )12( rrK .  Case III.  If 0,1,)( kxxf  i.e., in the presence of the absorbent term,  .)12(),(2txtxu  (18)   5Das et al.: Approx Analytical Solution of the Fractional Diffusion EquationPublished by Digital Commons @PVAMU, 2010106  Das et al. 4. Numerical Results and Discussion   In this article, the displacement ),( txu  for different fractional Brownian motions 32,21,31 and also for the standard case, the numerical values motion 1 are calculated Case II and Case III, for various values of time t  at 1x . The results thus obtained are depicted through Figure 1 for Case II and Figure 2 for Case III.  It is observed from Figure 1 that the displacement increases with time in the presence of both the external force and the reaction term but converges rapidly with the increase of .  For Brownian motion the rate of convergence is more and hence the effective damping is better than in standard motion. The rate of convergence is very high in comparison to the treatment carried out by Das and Gupta (2010).  Figure 2 shows the convergence of displacement ),( txu  in the presence of only the reaction term. It is obvious that the displacement converges to zero unlike the earlier case studies carried out by Das and Gupta (2010). It is seen that for the response to converge to zero, more time is required with the increase of . Therefore the effective damping is less as it goes to the Brownian motions from the standard motion.   5. Conclusion  There are three important goals that we have achieved through this study. First one is the usage of extremely simple, concise and highly efficient mathematical tool like HPM to solve the general fractional diffusion equation. Secondly, the effect of reaction term on the fractional diffusion equation with or without the presence of drift term (external force) has been analysed.  The most important part of this study is to compare the effect of reaction term on the fractional diffusion equation for different Brownian motions with the existing result of Das and Gupta (2010) while using different reaction term and the same absorbent term. Hence it is seen that the choice of this reaction term helps to exercise better damping on the dynamic response of the system. The results of the study have been clearly exhibited through graphs.   Plots reveal the rapid convergence of displacement ),( txu  with increase in  in the presence of both the external force and the reaction term. So, it can be concluded that dynamic performance and relative stability characteristics of a forced system are better under the influence of the new absorbent term. On the other hand dynamic performance and relative stability characteristics of an unforced system are better for Brownian motion than for standard motion. So we can infer that  should be chosen carefully keeping in consideration the performance requirements.  The authors strongly believe that the detailed study of the stability analysis will be beneficial and appealing to the scientists and engineers working in this field of research and they would be motivated to choose the proposed reaction term during their study. 6Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 9https://digitalcommons.pvamu.edu/aam/vol5/iss3/9AAM: Intern. J., Special Issue No. 1 (August 2010)  107   REFERENCES   Abbaoui, K. and Cherruault, Y. (1995). New ideas for proving convergence of decomposition methods, Comput. Math. Appl.Vol 29, pp 103-108. Belendez, A., Alvarez, M.L., Mendez, D.I., Fernandez, E., Yebra, M.S. and Belendez, T. (2008). Approximate Solutions for Conservative Nonlinear Oscillators by He's Homotopy Method, Z. Naturforsch. Vol 63a, pp 529-537.  Darvishi, M.T. and Khani, F. (2008). Application of He's Homotopy Perturbation Method to Stiff Systems of Ordinary Differential Equations, Z. Naturforsch. Vol 63a, pp 19-23.  Das, S. (2009). A Note on Fractional Diffusion Equations, Chaos, Solitons & Fractals. Vol 42, pp 2074-2079.  Das, S. and Gupta, P.K. (2010). An Approximate Analytical Solution of the Fractional Diffusion Equation with Absorbent Term and External Force by Homotopy Perturbation Method   Z. Fur Naturforsch. Sec. 65a, pp 182-190. He, J.H. (1999). Homotopy perturbation technique, Comp. Meth. Appl. Mech. & Engng. Vol 178, pp 257-262. He, J.H. (2000). A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non. Mech.Vol 35, pp 37-43. Mousa, M.M. and Ragab, S.F. (2008). Application of the Homotopy Perturbation Method to Linear and Nonlinear Schrödinger Equations, Z. Fur Naturforsch. Sec. A 63, 140-144.  Ray, S.S. and Bera, R.K. (2006). Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appld. Math. Comput. Vol 174, pp 329-336. Schot, A., Lenzi, M.K., Evangelista, L.R., Malacarne, L.C., Mendes, R.S. and Lenzi, E.K. (2007). Fractional diffusion equation with an absorbent term and a linear external force: Exact solution, Phys. Lett. Vol A 366, pp 346-350.           7Das et al.: Approx Analytical Solution of the Fractional Diffusion EquationPublished by Digital Commons @PVAMU, 2010108  Das et al.     0.2 0.4 0.6 0.8 150010001500200025003000        0.2 0.4 0.6 0.8 1102030405060                                                          (i)                                                                           (ii)        0.2 0.4 0.6 0.8 12.557.51012.515        0.2 0.4 0.6 0.8 123456                                                         (iii)                                                                           (iv)   Figure 1.    (i)  Plot of ),( txu  vs. t  at 1x  for ,31,1,1 k    (ii)  Plot  of  ),( txu vs. t  at 1x   for ,21,1,1 k   (iii) Plot of ),( txu  vs. t  at 1x  for ,32,1,1 k   (iv) Plot of ),( txu vs. t  at 1x  for 1,1,1 k .     t t t t ),( txu ),( txu ),( txu  ),( txu 8Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 9https://digitalcommons.pvamu.edu/aam/vol5/iss3/9AAM: Intern. J., Special Issue No. 1 (August 2010)  109     0.2 0.4 0.6 0.80.20.40.60.81           0.2 0.4 0.6 0.8 10.20.40.60.81                                                    (i)                                                                                   (ii)      0.2 0.4 0.6 0.8 10.20.40.60.81      0.2 0.4 0.6 0.8 1 1.2 1.40.20.40.60.81                                     (iii)                                                                                   (iv)   Figure 2.    (i) Plot of ),( txu vs. t  at 1x  for ,31,0,1 k (ii) Plot of ),( txu vs. t  at 1x  for ,21,0,1 k (iii) Plot of ),( txu  vs. t  at 1x  for ,32,0,1 k   (iv) Plot of ),( txu vs. t  at 1x  for 1,0,1 k  ),( txu ),( txu ),( txu  ),( txu t t t t 9Das et al.: Approx Analytical Solution of the Fractional Diffusion EquationPublished by Digital Commons @PVAMU, 2010

An Approximate Analytical Solution of the Fractional Diffusion Equation with External Force and Different Type of Absorbent Term - Revisited

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An Approximate Analytical Solution of the Fractional Diffusion Equation with External Force and Different Type of Absorbent Term - Revisited

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