Numerical solution for chemical kinetics system by using efficient iterative method

Transcription

1 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN Numerical solution for chemical kinetics system by using efficient iterative method M A AL-Jawary #1, R KRaham #2 #1 Head of Department of Mathematics, College of Education for Pure Science / Ibn AL-Haytham, Baghdad University, Iraq, Baghdad #2 Department of Mathematics, College of Education for Pure Science / Ibn AL-Haytham, Baghdad University, Iraq, Baghdad ABSTRACT: This paper presents the implementation of the new iterative method proposed by Daftardar-Gejji and Jafari (DJM) [V Daftardar-Gejji, H Jafari, An iterative method for solving non-linear functional equations, J Math Anal Appl 316 (2006) ] to solve a system of ordinary differential equations which represent of one of a mathematical models of a chemical kinetics problems In this iterative method the solution is obtained in the series form with easily computed components The results of the maximal error remainder values show that the present method is very effective and reliable The software used for the calculations in this study was Mathematica 9 Keywords: Chemical kinetics system, system of nonlinear differential equations, Numerical solution Iterative method Corresponding Author: MA AL-Jawary 1 INTRODUCTIO Many problems in physics, chemistry and biology can be represented by either linear or nonlinear ordinary or partial differential equations In chemistry for example, the chemical kinetics system is well described by nonlinear system of ordinary differential equations The Robertson is introduced this system in 1966 which represented the mathematical model of a chemical kinetics problem [1, 2] Several attempts have been made to develop analytic and approximate methods to solve this system Ganji et al[2] have successfully implemented both the Homotopy perturbation method (HPM) and Variation iteration method (VIM) to solve the system and obtained is approximate solutions Also, Khader [3] has applied the so called the Picard-pade technique which is based on a modification of the Picard iteration method (PIM) using Pade approximation Moreover, Matinfar et al [4] have implemented the homotopy analysis method (HAM) the solutions obtained using this method have high accuracy with respect to VIM and HPM presented in [2] This system of chemical kinetics was driven as a mathematical model, by the following [1, 3]: There are three spaces in the model of chemical process which are denoted by A, B and C, we can define the three reactions as: Page RS Publication, rspublicationhouse@gmailcom

2 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN A B (1) B+C A+C (2) B+B C (3) The concentrations of A,B and C can be denoted by u, v and w, respectively It is worth to assume the condensation of assembling 3 condensations in one, and that each of three constituent reactions will add to the condensation of any of the exactly at the expense of corresponding amounts of the reactions We will denoted to the reaction rate of Eq(1) is (a) this means that the rate at which v increases and at which u decreases, because of this reaction, will be equivalent to C operates as a catalyst in the production A from B and (b) represent reaction rate in the reaction Eq(2), meaning that the decrease of w and the increase of u in this reaction will be a rate equivalent to At last the rate at which this reaction will be equal to because the production of C from B will have constant rate equivalent to (c) The system of differential equations for the difference with time of the three condensations by putting all this components of the process together will be then [1,2,3,4]: with the initial conditions are: where a, b and c are the reaction rates Daftardar-Gejji and Jafari [5] have suggested a new iterative method for solving linear and nonlinear functional equations namely (DJM) The DJM has been successfully implemented by many authors for solving linear and nonlinear ordinary and partial differential equations of integer and fractional order, see [6 9] This method converges to the exact solution if it exists through successive approximations However, for concrete problems, a few number of approximations can be used for numerical purposes with high degree of accuracy The DJM does not require any restrictive assumptions for nonlinear terms as required by some existing techniques Recently, AL-Jawary et al [10 14] have successfully implemented The DJM for solving different linear and nonlinear ordinary and partial differential equations In this work we will implemented the DJM to solve a system of chemical kinetics of nonlinear ordinary differential equations This paper has been organized as follows: In section 2, the DJM is introduced and discussed In section 3, solving the system of chemical kinetics by the DJM will be given In section 4, numerical simulation is illustrated and discussed Finally, in section 5, the conclusion is given 2 AN ITERATIVE METHOD (DJM) Let us consider the following general functional equation [5]: Page RS Publication, rspublicationhouse@gmailcom

3 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN y=n(y)+f (7) Where N is a nonlinear operator and f is a known function The solution y have the series form: The nonlinear operator N can be decomposed as From Eqs (8) and (9), the Eq(7) is equivalent to The recurrence relation can be defined as:, m=1,2,3, (11) Then, m=1,2, (12) then For the convergence of DJM, we refer the reader to [15] 3 SOLVING THE SYSTEM OF CHEMICAL KINETICS BY DJM Let us rewrite the Eqs (4), (5) and (6) as: Page RS Publication, rspublicationhouse@gmailcom

4 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN and the initial conditions are By integrating both sides of the Eqs (14), (15)and (16) once from 0 to t and using the intial conditions,the following equations will be obtained: We define the nonlinear operators, and by : From equations (20), (21) and (22), then the equations (17), (18) and (19) can be written as: The solution u and v and w take the series form By using Eq(26) the nonlinear operators, and can be decompose as Page RS Publication, rspublicationhouse@gmailcom

5 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN By substituting Eqs (27)-(29) in Eqs (23), (24) and (25), we obtain The following approximations are achieved as:,,, 1) ( 0+ 1), Page RS Publication, rspublicationhouse@gmailcom

6 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN , and so on, we will get successive approximations of and and Our calculations are obtained by Mathematica 9 For a study the accuracy of the achieved approximate solution, Relevance functions of the error remainder will be [16, 17], and the maximal error remainder parameters are 4 NUMARICAL SIMULATIONS: We assume that the values of the three reaction rates are: a=01, b=002and c=0009, exactly the same values as in [1, 3] In order to assess the convergent of DJM for the system of chemical kinetics the error remainders and the maximal error remainders have been computed It can be seen clearly from Table 1 the values of MER 1,n,MER 2,n and MER 3,n for the system are decrease when the number of iterations are increase Table1: The MER 1,n,MER 2,n and MER 3,n for the system (31) (32) (33) n MER 1,n MER 2,n MER 3,n Moreover, Figures 1,2 and 3 show the analysis of the maximal error remainders for MER 1,n, MER 2,n and MER 3,n respectively, where the points are lay on a straight lines which mean we achieved exponential rate of convergence Page RS Publication, rspublicationhouse@gmailcom

7 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN Fig1: Logarithmic plots of MER 1,n against n is 1 through 4 Fig2: Logarithmic plots of MER 2,n against n is 1 through 4 Fig2: Logarithmic plots of MER 3,n against n is 1 through 4 Page RS Publication, rspublicationhouse@gmailcom

8 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN CONCLUTION In current paper, the DJM has been successfully applied to solve a system of ordinary differential equations which represent of one of a mathematical models of a chemical kinetics problems The DJM provides the solutions in the form of convergent series with easily computed components It is economical in terms of computer power/memory and does not involve tedious calculations without any restricted assumption and it is seems that the DJM appears to be very accurate to employ with reliable results It has been achieved that from figures and table that the maximal error remainders decreased when the number of iterations are increased REFRENCES [1] H Aminikhah, An analytical approximation to the solution of chemical kinetics system, Journal of King Saud University-Science, Vol 23, pp , 2011 [2] D D Ganji, M Nourollahi, E Mohseni, Application of He's methods to nonlinear chemistry problems, Computers and Mathematics with Aplications, Vol 54, pp , 2007 [3] M M Khader, On the numerical solutions for chemical kinetics system using picard-pade technique, Journal of King Saud University-Science, Vol 25, pp , 2013 [4] M Matinfar, M Saeidy, B Gharahsulfu, M Eslami, Solutions of nonlinear chemistry problems by homotopy analysis method, Compurtational Mathematics and Modeling, Vol 25(1), pp , 2014 [5] V Daftardar-Gejji, H Jafari An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, Vol 316, pp , 2006 [6] S Bhalekar, V Daftardar-Gejji, Solving a system of nonlinear functional equations using revised new iterative method, World Academy of Science, Engineering and Technology, Vol 68, pp 08-21, 2012 [7] S Bhalekar, V Daftardar-Gejji, New iterative method: Application to partial differential equations, Applied Mathematics and Computation, Vol 203, pp , 2008 [8] V Daftardar-Gejji, S Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Computers and Mathematics with Applications, Vol 59, pp , 2010 [9] M Yaseen, M Samraiz, S Naheed, The DJ method for exact solutions of Laplace equation, Results in Physics, Vol 3, pp 38-40, 2013 [10] MA AL-Jawary, A reliable iterative method for solving the epidemic model and the prey and predator problems, International Journal of Basic and Applied Sciences, Vol 3, pp , 2014 [11] MA AL-Jawary, Approximate solution of a model describing biological species living together using a new iterative method, International Journal of Applied Mathematical Research, Vol 3, pp , 2014 [12] MA AL-Jawary, A reliable iterative method for Cauchy problems, Mathematical Theory and Modeling, Vol 4, pp , 2014 [13] MA AL-Jawary, Exact solutions to linear and nonlinear wave and diffusion equations, International Journal of Applied Mathematical Research, Vol4, pp , 2015 [14] MA AL-Jawary, H R AL-Qaissy, A reliable iterative method for solving Volterra integro-differential equations and some applications for the Lane-Emden equations of the first kind, Monthly Notices of the Royal Astronomical Society, Vol 448, pp , 2015 Page RS Publication, rspublicationhouse@gmailcom

9 International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan Feb 2016 Available online on ISSN [15] S Bhalekar, V Daftardar-Gejji, Convergence of the new iterative method, International Journal of Differential Equations, Vol 2011(Article ID ), 10 pages, 2011 [16] JDuan, R Rach, AM Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method Journal of Mathematical Chemistry, Vol 53, pp , 2015 [17] MA AL-Jawary, G H Radhi, The Variational iteration method for calculating carbon dioxide absorbed into phenyl glycidyl ether IOSR Journal of Mathematics, Vol 11, pp , 2015 Page RS Publication, rspublicationhouse@gmailcom