Symmetry and Numerical Solutions for Systems of Non-linear Reaction Diffusion Equations

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1 Symmery and Numerical Soluions for Sysems of Non-linear Reacion Diffusion Equaions Sanjeev Kumar* and Ravendra Singh Deparmen of Mahemaics, (Dr. B. R. Ambedkar niversiy, Agra), I. B. S. Khandari, Agra-8 Id: and Absrac: Many imporan applicaions are available for nonlinear reacion-diffusion equaion especially in he area of biology and engineering. Therefore a mahemaical model for Lie symmery reducion of sysem of nonlinear reacion-diffusion equaion wih respec o one-dimensional Algebra is carried ou in his work. Some classes of analyical and numerical soluions are obained and expressed using suiable graphs.. Inroducion: Transpors of molecular oxygen from he blood plasma o he living issue of he skeleal muscle or brain across he capillary walls are nowadays very imporan opic. Several quesions arise in our mind like (i) wha facors affec he supply of oxygen issue cell respiraions? (ii) wha happens when we inhale oxygen a low concenraion (iii) wha is he influence of axial and redial diffusion of oxygen in blood, oxygen diffusiviy in issue ec. Whenever we alk abou he problem of diffusion, parallel we have o discuss abou he diffusion or diffusion reacion equaion. Diffusion reacion equaions are hen bifurcaed ino wo ways, linear and nonlinear diffusion reacion equaions. This problem is o obain some differen soluions wih respec o a coupled nonlinear reacion diffusion equaions. Coupled sysems of nonlinear diffusion equaions have many imporan applicaions in mahemaical physics, chemisry and biology. In populaion biology, he reacion erms models growh and he diffusion erm accouns for migraion. The classical diffusion erm originaes from a model in physics. Recen research indicaes ha he classical diffusion equaion is inadequae o model many real siuaions, where a paricle plume spreads faser han he classical model predaes and may exhibi significan asymmery. This siuaion is called anomalous diffusion. Now as far as he soluion of such equaions is concern, we are having several approaches. Cherniha e al [6] worked on Lie symmery of nonlinear mulidimensional diffusion equaion. Alhough symmery is very much imporan herefore Nikiin e al [8] worked on he sysem of reacion diffusion equaions and heir symmery properies. Teyana [] discussed abou boh symmery and soluion for sysem of nonlinear reacion diffusion equaions. Lo of applicaions are available for such equaions, Sharma e al [5] worked on he effec of nonzero bulk flow and non-mixing on diffusion wih variable ransfer of solue while Cenral e al

2 [] discussed abou he spaial ecology via reacion-diffusion equaions, while Kumar e al [] worked on a compuaional sudy of oxygen ranspor in he body of living organism.. Mahemaical Model: In he presen model we consider a general sysem of nonlinear diffusion equaion of he following form ( a a ) F. (a) ( a a ) G. (b) where a, a, a, a, are consan parameer wih relaion a a - a a, and F,G are he funcion of and, while, are he funcions dependen on and x. An invesigaion of he equaions () can be underaken wihin he framework of he classical Lie algorihm which reduces he problem o deermine he exac and numerical soluion. The classical Lie symmeries of parial differenial equaions and classical symmeries of sysems of wo nonlinear diffusion equaions wih n+ independen variable,x,...,x n were described. All possible non-linearies F, G and he corresponding group generaors were found. In he presen work, using he resuls obained in [7], we carried ou symmery reducion of equaion () wih respec o one-dimensional symmery algebras and F, G are defined up o arbirary funcions. 3. Symmery Reducions: If and are arbirary funcion of and a =a =a, a =, a =b, hen a sysem of ype () may be considered as: a exp ( ) (a) b exp ( ) ( ) (b) Now if he Greek leers, R denoe he arbirary coefficien, and D i, X, B are various ype of dilaaion and special ransformaion generaor, hen some of he operaors consider for his model, are as follows: X, D B, B bijub u a D x n, D3 x p u (3) where b ij are he elemens of he * marix B, and, n are parameers used in he definiions of nonlinear erm, and B, which have been specified by B (4)

3 The sysem of () permis he following symmery operaor: X X D, (5) Now using Lie algorihm, he corresponding soluion is: w ( z ) (6a) In( vx ) w ( z ) (6b) z ( vx v ). (6c) Now if we subsiue he soluion (3) ino (), hen we come o he following reduced equaions, which is a sysem of ordinary differenial equaion. az vz 4v z 8v exp ( ) (7a) v b v bz 4v bz exp ( ) ( ) vz (7b) 4. Numerical Soluions: Consider he following sysem () for he numerical soluion a b x x exp ( ) exp ( ) ( ) We solve he above parial differenial equaion hrough he MATLAB 6., and consider following value of he represen he parameers: a, and b, 5.73, (8) We have aken x= o and = o sec. Figure () and () represens he value of and respecively wih respec o ime and disance x.

4 shows he funcion a differen value of disances x and ime The value of is decreasing wih respec o ime and disance x. shows he funcion a differen value of disances x and ime The value of is decreasing wih respec o ime and disance x.

5 5. Condiional symmery and exac soluion: Thus we presened reducion of equaion () using heir classical symmery found in [7]. In his secion we presen exac soluion of equaion () found by condiional symmery reducion. 3 (9a) 3 (9b) where and are funcion of. Condiional symmery operaor: X 3 3 k x ( x k). () The ansaz ( x k ) ( z) (a) ( x k ) ( z) (b) z x kx 3 (c) Reduce equaion (4) o he following sysem: 3 (b) 3 (b) where and are funcion of. Depending on he form of he funcions,, we receive differen soluion he sysem. ) = a >, = b <, where a and b are consans. a ( x, ) ( x k ) sd x kx a 3 ; b ( x, ) ( x k ) ds x kx b ), a 3 ; a ( x, ) ( x k ) sd x kx a 3 ; (3b) (3b) (4b) ( x, ) ( x k) x kx 3; C C. (4b)

6 6. Conclusions: Through secion 4, we have he numerical resuls for he and a differen values of disances x and ime. The funcion is decreasing wih respec o ime and disance x and he funcion is increasing wih respec o ime and disance x. In work paper we have presened a pos processing algorihm ha he sysem of nonlinear reacion diffusion equaion solvable in exac and numerical form. Such ype of linear and nonlinear reacion, diffusion, and reacion-diffusion equaions may be used o model spaio-emporal processes in physical sysems, like asrophysical fusion plasmas. We sugges he use of such models or sysems o describe he naural srucures as paerns in chemisry, ecology, biology and physics or he signal behavior in he neural conex References:. Kapoor, J. N. (985); Mahemaical models in biology and medicine, Aff. Eas-Wes Press, New Delhi.. Brion N.F. (986); Reacion-diffusion and heir applicaions o biology, Academic Press. Inc., Harcour Brace Jovanovich Publishers, London. 3. Cherniha R. (996); A consrucive mehod for consrucion of new exac soluions of nonlinear evoluion equaions, Rep. Mah. Phys., ol.38, Grindord P. (996); Theory and applicaions of reacion-diffusion equaions, second edu. Oxford Applied Mahemaical and Compuing Science Series. The Clarendon Press Oxford niversiy Press, New York. 5. Sharma G.C., Kumar S., Jain M. and Saral R. N. (998); The effec of nonzero bulk flow and non-mixing on diffusion wih variable ransfer of solue, In. J. Appl. Sci. Compu. ol. 4, 3, Cherniha R. and King J. R. (); Lie symmery of nonlinear mulidimensional reaciondiffusion sysem, J. Phys. A, ol.33, Nikiin A. G. and Wilshire R.J. (); Symmery of sysem of nonlinear reaciondiffusion equaions, Proceedings of Third Inernaional Conference, Symmery Mahemaical Physics, ol.3, Par, Nikiin A.G. and Wilshire R. J. (); sysem of reacion-diffusion equaion and heir symmery properies, J. Mah. Phys., ol.4, N 4, Bird R.B., Sewar W.E. and Lighfoo E.W. (); Transpor Phenomena, John Wiley & Sons (Asia) Pe. Ld, India.. Teyana B. (); Symmery and exac soluions for sysems of nonlinear reacion-diffusion equaions, Symmery Mahemaical Physics ol.43, Par, Cenral R.S. and Cosner C. (3); Spaial ecology via reacion-diffusion equaions, Wiley series in Mahemaical and Compuaion Biology, John Wiley & Sons Ld. Chircheser.. Kumar, S. and Kumar, N. (5); A compuaional sudy of oxygen ranspor in he body of living organism, In. Jour. Engg., 8, 4,

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