Electronic Journal of Mathematical Analysis and Applications, Vol. 4(1) Jan 2016, pp. 197-204. ISSN: 2090-729(online) http://fcag-egypt.com/Journals/EJMAA/ ———————————————————————————————— COMPOSITE FINITE DIFFERENCE SCHEME APPLIED TO SOME NONLINEAR EVOLUTION EQUATIONS S. A. EL MORSY Abstract. This paper is concerned with the application of the composite finite difference scheme (CFDS) to some classes of evolution equations. Three models of evolution equations are studied. The first model is the nonlinear reaction-diffusions equation (NRD) with a reaction term, while the second is modified Korteweg de Vries equation (mKdV) and the third is the Fitzhugh?Nagumo equation (FN). Numerical examples showed that the CFDS give high accuracy. 1. Introduction Nonlinear evolution equations are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, etc. The study of nonlinear partial differential equations is very important. Many methods, exact, approximate and purely numerical are available for solution of nonlinear partial differential equations [1]-[30]. Reaction-diffusion equations (RD) are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. A great deal of research work has been published on the development of numerical and analytical solutions of NRD equations [1]-[10]. In recent years, many physicists and mathematicians have paid much attention to the Fitzhugh? Nagumo (FN) equation due to its importance in mathematical physics. The Fitzhugh?Nagumo equation has various applications in the fields of flame propagation, logistic population growth, neurophysiology, branching Brownian motion process, autocatalytic chemical reaction and nuclear reactor theory; see, e.g. [11]-[16]. This equation is an important nonlinear reaction diffusion equation and usually used to model the transmission of nerve impulses [16]. Numerical schemes for FN equations [17]-[19] by collocation method and the ?Hopscotch? finite difference scheme first proposed by Gordon [19], and further developed by Gourlay [20]-[21]. 2010 Mathematics Subject Classification. 34A12. Key words and phrases. Partial differential equations, nonlinear reaction-diffusions equation, modified Korteweg de Vries equation and Fitzhugh?Nagumo equation. Submitted April 2, 2014. Revised April 22, 2014. 197 198 S. A. EL MORSY EJMAA-2016/4(1) A great deal of research work has been invested during the past decades in the study of the mKdV equation [22]-[30]. The main goal of these studies was its analyticalzz and numerical solutions. Several different approaches, such as Backland transformation, a bilinear form, and a Lax pair, have been used independently, by which Anjan et al. [22]-[25] obtain soliton and multi-soliton solutions for this equation. The aim of this paper is to apply the CFDS to obtain the solutions for the three different types of nonlinear partial differential equations such as, nonlinear Reaction?diffusion equation (NRD), Fitzhugh? Nagumo (FN) and modified Kortewege de Vries equation (mKdV) which are important equations. 2. The nonlinear reaction-diffusion equation with reaction term An example of practical interest is known as the nonlinear reaction-diffusions equation (NRD) with a reaction term [7, 8].this equation takes the form [8]. ut − u2xx = pu − qu2 , (x, t) ∋ QT (1) Here QT = Ω × I, Ω ≡ (a, b), I = (0, T ), a and b are real positive constants. We consider equation (1) associated with initial condition u(x, 0) = u0 (x). In Finite difference method (FDM) the domain is discretized to a finite number of points forming a mesh with horizontal step size h = b−a N , N is the number of intervals, 0 ≺ i ≼ N and k is the time step such that T = k ∗j, 0 ≼ j ≼ M . The derivatives are replaced by difference formulas [31]-[32] as follows, for i = 1, 2 we use the forward formula −3uji + 4uji+1 − uji+2 (ux )ji = 2h (uxx )ji = 2uji − 5uji+1 + 4uji+2 − uji+3 h2 (2) −5uji + 18uji+1 − 24uji+2 + 14uji+3 − 3uji+4 2h3 while for i = 3, N − 2 we use the central formulas (uxxx )ji = (ux )ji = (uxx )ji = uji+1 − uji−1 2h uji+1 − 2uji − uji−1 2h2 (3) uji+2 − 2uji+1 + 2uji−1 − uji−2 2h3 And for i = N − 1, N we use the backward formulas (uxxx )ji = (ux )ji = (uxx )ji = (uxxx )ji = 3uji − 4uji−1 + uji−2 2h 2uji − 5uji−1 + 4uji−2 − uji−3 h2 5uji − 18uji−1 + 24uji−2 − 14uji−3 + 3uji−4 2h3 (4) EJMAA-2016/4(1) COMPOSITE FINITE DIFFERENCE 199 3. Application of CFDS to the nonlinear reaction-diffusion equation Consider the nonlinear PDE (1), we can rewrite: ut = u2xx + pu − qu2 multiply both sides of (5) by ∂F ∂u (5) , we have ∂F ∂u ∂F 2 = (u + pu − qu2 ) ∂u ∂t ∂u xx (6) or ∂F 2 ∂F = (u + pu − qu2 ) (7) ∂t ∂u xx where F , is any continuous and differentiable function. If we choose F (u) = ln u, we obtain the Exponential finite difference method (Exp. FDM) [33]-[36]. The Logarithmic finite difference method (Log. FDM) [32] is obtained when we set F (u) = exp u. 3.1. Exponential finite difference method applied to the nonlinear reactiondiffusion equation. In this sub-section we will apply the Exponential finite difference method to (7). The usual forward difference formula leads to, ∂F ∂t = F (uj+1 )−F (uji ) i k is the time step. Substitute in (7)), we have F (uj+1 ) − F (uji ) ∂F 2 i = (u + pu − q ∗ u2 ) k ∂u xx (8) or ∂F 2 (u + pu − q ∗ u2 )) ∂u xx setting F (u) = ln u , then (9) takes the form, F (uj+1 ) = F (uji ) + k( i ln uj+1 = ln uji + i k uji ((uji )2xx + puji − q(uji )2 ) (9) (10) so, uj+1 = exp( i k uji ((uji )2xx + puji − q(uji )2 ) (11) Applying the difference formulas (2)-(4) to (11), we have the following recurrence relations: uj+1 = (uji ) exp( i j j j j k 2(ui )2 − 5(ui+1 )2 + 4(ui+2 )2 − (ui+3 )2 +p(uji )−q(uji )2 ), i = 1, 2 h2 uji (12) uj+1 = (uji ) exp( i uj+1 = (uji ) exp( i j j j k (ui+1 )2 − 2(ui )2 + (ui−1 )2 +p(uji )−q∗(uji )2 ), i = 3 : N −2 (13) j 2 2h ui j j j j k 2(ui )2 − 5(ui−1 )2 + 4(ui−2 )2 − (ui−3 )2 +p(uji )−q∗(uji )2 ), i = N −1, N h2 uji (14) 200 S. A. EL MORSY EJMAA-2016/4(1) 3.2. Logarithmic finite difference method applied to the nonlinear reactiondiffusion equation. In (Log. FDM) we assume F (u) = exp u, equation (9) transformed to exp(uj+1 ) = exp(uji ) + k exp(uji )((uji )2xx + puji − q(uji )2 ) i (15) uj+1 = uji + ln(1 + k((uji )2xx + puji − q(uji )2 ) i (16) we have, Similarly applying the difference formulas (2)-(4) to (16), we have the following recurrence relations: uj+1 = uji +ln(1+k( i (2(uji )2 − 5(uji+1 )2 + 4(uji+2 )2 − (uji+3 )2 ) +puji −q(uji )2 )), i = 1, 2 h2 (17) uj+1 = uji +ln(1+k( i uj+1 = uji +ln(1+k( i (uji+1 )2 − 2(uji )2 + (uji−1 )2 +puji −q(uji )2 )), i = 3 : N −2 (18) 2h2 (2(uji )2 − 5(uji−1 )2 + 4(uji−2 )2 − (uji−3 )2 ) +puji −q(uji )2 )), i = N −1, N h2 (19) 4. Fitzhugh?Nagumo equation The classical Fitzhugh?Nagumo equation [16]-[17],is given by ut = uxx + u(1 − u)(p − u) (20) where 0 ≼ p ≼ 1 and u(x, t) is the unknown function depending on the temporal variable t and the spatial variable x. This equation combines diffusion, and nonlinearity which is controlled by the term u(1 − u)(p − u). When p = 1, (20) reduces to the real Newell?Whitehead equation. To apply CFDS, we reset (20) as follows, F (uj+1 ) − F (uji ) ∂F i = (uxx + u(1 − u)(p − u)) k ∂u (21) ∂F (uxx + u(1 − u)(p − u)) ∂u (22) ie. F (uj+1 ) = F (uji ) + k i 4.1. Exp. FDM method applied to FN equation. Replacing the derivatives in (22) by the difference formulas (2)-(4), we obtain uj+1 = (uji ∗exp i j j j j k 2ui − 5ui+1 + 4ui+2 − ui+3 ( +uji (1−uji )(p−uji )), i = 1, 2 (23) h2 uji uj+1 = (uji ∗ exp i j j j k ui+1 − 2ui + ui−1 ( + uji (1 − uji )(p − uji )), i = 3, N − 2 (24) 2h2 uji EJMAA-2016/4(1) COMPOSITE FINITE DIFFERENCE 201 and uj+1 = (uji ∗ exp i j j j j k 2ui − 5ui−1 + 4ui−2 − ui−3 ( + uji (1 − uji )(p − uji )), i = N − 1, N h2 uji (25) 4.2. Log. FDM method applied to FN equation. For the Log. FDM, we have the following iterative formulas, uj+1 = (uji + ln(1 + k( i 2uji − 5uji+1 + 4uji+2 − uji+3 + uji (1 − uji )(p − uji ))), i = 1, 2 h2 (26) uj+1 = (uji + ln(1 + k( i uji+1 − 2uji + uji−1 + uji (1 − uji )(p − uji ))), i = 3, N − 2 (27) 2h2 and and for i = N − 1, N we have uj+1 = uji + ln(1 + k( i 2uji − 5uji−1 + 4uji−2 − uji−3 + uji (1 − uji )(p − uji )) h2 (28) 5. Application of CFDS to the modified KdV equation Consider the modified Korteweg?de Vries equation (mKdV), which takes the form ut + 6u2 ux + uxxx = 0 (29) F (uj+1 ) = F (uji ) − k(6(uji )2 (uji )x + (uji )xxx ) i (30) Simliarly, we obtain 5.1. Exp. FDM method applied to mKdv equation. The recurrence relations of Exp. FDM are, uj+1 = (uji ∗exp i uj+1 = (uji ∗exp i −k uji −k uji (6(uji )2 −3uji + 4uji+1 − uji+2 −5uji + 18uji+1 − 24uji+2 + 14uji+3 − 3uji+4 + ), i = 1, 2 2h 2h3 (31) (6(uji )2 uji+1 − uji−1 uji+2 − 2uji+1 + 2uji−1 − uji−2 + ), i = 3, N −2 2h 2h3 (32) (6(uji )2 3uji − 4uji−1 + uji−2 5uji − 18uji−1 + 24uji−2 − 14uji−3 + 3uji−4 + ), i = N −1, N 2h 2h3 (33) and uj+1 = (uji ∗exp i −k uji 202 S. A. EL MORSY EJMAA-2016/4(1) Table 1. Numerical results of solving NRD equation at different times,absolute errors for CFDS x 2 3 4 5 6 7 8 9 10 11 12 t = 0.01 0.0001351 0.0000716 0.0000363 0.0000695 0.0000416 0.0000249 0.0000150 0.0000091 0.0000055 0.0000028 0.0000017 t = 0.1 0.0011230 0.0008931 0.0002172 0.0006197 0.0003921 0.0002358 0.0001420 0.0000857 0.0000516 0.0000258 0.0000184 t = 0.5 0.0304806 0.0159891 0.0016654 0.0016458 0.0014175 0.0009075 0.0005527 0.0003337 0.0001966 0.0000844 0.0001753 5.2. Log. FDM method applied to mKdV equation. In case of Log. FDM equation (30) transformed to uj+1 = uji + ln(1 − k(6(uji )2 (uji )x + (uji )xxx )) i (34) Similarly applying the difference formulas (2)-(4) to (34), we have the following recurrence relations: uj+1 = uji +ln(1−k(6(uji )2 i −3uji + 4uji+1 − uji+2 −5uji + 18uji+1 − 24uji+2 + 14uji+3 − 3uji+4 + )), i = 1, 2 2h 2h3 (35) uj+1 = uji +ln(1−k(6(uji )2 i uji+1 − uji−1 uji+2 − 2uji+1 + 2uji−1 − uji−2 + )), i = 3, N −2 2h 2h3 (36) and uj+1 = uji +ln(1−k(6(uji )2 i 3uji − 4uji−1 + uji−2 5uji − 18uji−1 + 24uji−2 − 14uji−3 + 3uji−4 + )), i = N −1, N 2h 2h3 (37) 6. Numerical examples In this section, we apply preceding algorithm to three numerical examples associated with the appropriate initial conditions . Example.1 consider the reaction-diffusions equation (1), when 2 ≼ x ≼ 12 in case of p = 1, q = 1, at h = 1and k = 0.00001, we start with the initial approximation, −x x+t u(x, 0) = 13 (3 + e 2 ). The exact solution is u(x, t) = 31 (3 + e 2 ). . Example.2 consider the mKdV equation (29), when 10 ≼ x ≼ 20 in case of h = 1and k = 0.00001, we start with the initial approximation, u(x, 0) = sech(x). The exact solution is u(x, t) = sech(x − t). EJMAA-2016/4(1) COMPOSITE FINITE DIFFERENCE 203 Table 2. Numerical results of solving mKdV equation at different times x 10 11 12 13 14 15 16 17 18 19 20 t = 0.01 4.7 ∗ 10−7 1.7 ∗ 10−7 6.4 ∗ 10−8 1.2 ∗ 10−8 4.4 ∗ 10−9 1.7 ∗ 10−9 6.3 ∗ 10−10 2.3 ∗ 10−10 8.1 ∗ 10−11 1.0 ∗ 10−9 4.7 ∗ 10−10 t = 0.1 5.2 ∗ 10−6 1.6 ∗ 10−6 4.9 ∗ 10−7 1.3 ∗ 10−7 2.5 ∗ 10−8 2.0 ∗ 10−8 6.5 ∗ 10−9 2.8 ∗ 10−9 5.8 ∗ 10−10 9.8 ∗ 10−9 7.5 ∗ 10−9 t = 0.5 0.00009056 0.00001313 0.00001414 2.8 ∗ 10−6 1.1 ∗ 10−6 3.9 ∗ 10−7 5.6 ∗ 10−8 5.3 ∗ 10−8 1.6 ∗ 10−8 1.8 ∗ 10−7 1.3 ∗ 10−7 Table 3. Numerical results of solving FN equation at different times x 0 1 2 3 4 5 6 7 8 9 10 t = 0.01 0.0002393 0.0001758 0.0000301 0.0000025 0.0000034 0.0000036 0.0000024 0.0000013 0.0000007 0.0000086 0.0000042 t = 0.1 0.0022600 0.0018887 0.0003588 0.0000086 0.0000312 0.0000350 0.0000237 0.0000133 0.0000027 0.0000931 0.0000238 t = 0.5 0.0045646 0.0126318 0.0032443 0.0004057 0.0001697 0.0001613 0.0001158 0.0000565 0.0000676 0.0006180 0.0005093 Example.3 consider the Fitzhugh?Nagumo equation (20), when 0 ≼ x ≼ 10 in case of h = 1and k = 0.00001, we start with the initial approximation, u(x, 0) = 2p−1 1 x 1 x √ √ 2 (1 + tanh 2 2 ). The exact solution is u(x, t) = 2 (1 + tanh( 2 2 − 4 )), p = 0.75. . Tables (1- 3) illustrate the numerical results of solving RD equation, mKdV equation and FN equation using CFDS at different times. 7. Conclusion The CFDS is effective for solving linear and nonlinear partial differential equations especially for small time intervals. 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