PROPERTIES OF SELF SIMILAR SOLUTIONS OF REACTION-DIFFUSION SYSTEMS OF QUASILINEAR EQUATIONS D. K. MUHAMEDIYEVA

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1 Iteratoal Joural of Mechacal ad Producto Egeerg Research ad Developet (IJMPERD) ISSN (P): ; ISSN (E): 49-8 Vol. 8, Issue, Apr 8, TJPRC Pvt. Ltd PROPERTIES OF SELF SIMILAR SOLUTIONS OF REACTION-DIFFUSION SYSTEMS OF QUASILINEAR EQUATIONS ABSTRACT D. K. MUHAMEDIYEVA Research Scholar Tashket Uversty of Iforato Techologes Naed after Muhaad al-khwarz Scetfc ad Iovato Ceter of Iforato ad Coucato Techologes, Tashket, Uzbeksta I ths paper, cosdered a parabolc syste of two quas lear reacto-dffuso equatos wth the source ad absorpto task ad the propertes of self-slar solutos of a syste of quas lear reacto-dffuso equatos for the source ad absorpto task. Self-slar syste of equatos s costructed by the ethod of olear splttg. Estates of the solutos ad the free boudary that arses ths case are foud, whch akes t possble to choose sutable tal approxatos for each value of the uercal paraeters. KEYWORDS: Source ad Absorpto Equato, Parabolc Syste, Quas Lear Equatos, Self-Slar Syste of Equatos & Reacto-Dffuso Receved: Ja, 8; Accepted: Ja 3, 8; Publshed: Mar 8, 8; Paper Id.: IJMPERDAPR864 INTRODUCTION Study of olear atheatcal odels of varous physcal, bologcal, checal ad other pheoea ad processes s oe of the portat drectos of atheatcal odelg. As exaples, we ote such physcal theores as o-lear quatu echacs, olear electrodyacs ad optcs, o-lear theory of plasas, olear acoustcs, olear coducto, olear dffuso, ad other theores based o atheatcal odels whch are olear dfferetal equatos partal dervatves. The study of the lear atheatcal odels of physcal processes are easy to study, sce the uderlyg lear dfferetal equatos developed geeral ethods for ther soluto [,]. I appled tasks, the actual physcal processes are olear ad for ther adequate descrpto, oe should use olear atheatcal odels. Orgal Artcle Nolear odels of atheatcal physcs, descrbg pheoea ad processes a wder rage of physcal paraeter chages ad have a better capacty of forato about these pheoea ad processes. Lear odels are typcally specal cases of olear odels. They ca gve oly a approxate pcture of the pheoeo uder study wthout detfyg the observed effects. Studes show that the oleartes chage ot oly the quattatve characterstcs of the processes, but the qualtatve pcture of ther behavor. Iterestgly, fro the pot of vew of applcatos to study the followg classes of olear dfferetal equatos whch the ukow fucto ad the dervatve of ths fucto cossts of expoetal way. The, wth the coparso theores of solutos of ths class ca be exteded. These types of oleartes are ofte ecoutered probles of the theory of fltrato, dffuso, theral coductvty, agetohydrodyacs, bologcal populatos, the ol dustry, etc. [3,4,5,6]. edtor@tjprc.org

2 556 D. K. Muhaedyeva These odels ore accurately descrbe the physcs of the process ad, therefore, ther research shows that there are ew effects related to the olearty of the studed process. So was foud the effects of the fte speed of propagato of perturbatos [], localzato of solutos ad dfferet odes of processes. The frst effect of fte speed of propagato of perturbatos was obtaed ad appled to the proble of olear theral coductvty, the work of Zeldovch ad A. S. Kopaeetz[], for the proble of olear flterg the work of G. I. Bareblatt[,3], whch are depedetly fro each other, ad got ths effect. Further atheatcal questos were studed works of O. A. Olek, A. S. Kalashkov [5,6,7], A. A. Saarsky ad hs studets [,3-4], B. Kerr [3]. Stateet of the Proble I the feld Q{(t,: <t<, x R} cosders a parabolc syste of two quaslear equatos of source ad absorpto u σ u Du + k( t) u ( u ), t x x u σ u Du + k( t) u ( u ), t x x () u ( t u, u t u(, () Whch descrbes the process of heat dstrbuto a olear two-copoet evroet, the dffuso coeffcet of whch s equal to u u t, - desred soluto. ( σ D u ad σ σ, σ - postve real ubers, u t,,, D u, u, ( Estates of solutos are foud, whch akes t possble to choose sutable tal approxatos for each value of uercal paraeters. Costructo of a Self-Slar Syste of Equatos Self-slar syste of equatos s costructed by the ethod of olear splttg[4]. d σ df ξ df ( f ) + + µ ( f f f ), dξ dξ dξ d σ df ξ df ( f ) + + µ ( f y f ), dξ dξ dξ where µ ( γ σ ), To costruct the upper soluto of the syste () - (), we troduce fuctos f ( ξ ) A( a ξ ), f ( ξ ) B( a ξ ), + +, where σ, ( b) ax(, b) σ +, Ipact Factor (JCC): NAAS Ratg: 3.

3 Propertes of Self Slar Solutos of Reacto-Dffuso Systes of Quaslear Equatos 557 t k ( η ) dη + u ( t, u ( t, e f ( ξ ), t k ( η ) dη + u ( t, u ( t, e f ( ξ ), f ( ξ ), f ( ξ ) и τ ( t) -the fuctos defed above. Cross-Dffuso Syste of Two Quaslear Reacto-Dffuso Equatos of the Source ad Absorpto I the doa Q{(t,: <t<, x R} cosdered a parabolc syste of two quaslear reacto-dffuso equatos for the source ad absorpto u σ u Du + k( t) u ( u ), t x x u σ u Du + k( t) u ( u ), t x x (3) u t u (, u t u (, whch descrbes the process of process of absorpto a olear two-copoet evroet, dffuso coeffcet whose are σ D u ad u u t, - sought solutos. ( σ σ, σ - postve real ubers, u t,,, D u, u, ( assug (3) the case 3 Let γ σ, > γ σ γ σ, σ b + ) + ( b + ) σ ( b + ) + ( b ), c > w ( τ ( t), f ( ξ ), ξ x / τ,,, ( + / ad takg to accout that the equato for w γ σ, obta the syste ( τ,. I ths case, wthout lower ters always have a self-slar soluto d σ df ξ df ( f ) + + µ ( f f f ), dξ dξ dξ d σ df ξ df ( f ) + + µ ( f f f ), dξ dξ dξ (4) where µ. γ σ 3 To costruct the upper soluto of the syste (4), we troduce fuctos edtor@tjprc.org

4 558 D. K. Muhaedyeva where f ( ξ ) A( a ξ ), f ( ξ ) B( a ξ ), + + ( σ + ), ( σ + )( σ + ) ( σ + ), ( b) ax(, b) ( σ + )( σ + ) +, σ df σ+ f B γ ξ f C(, ), dξ σ df σ + f A γξ f C(, ) dξ ad d σ df σ + df f γ B f + ξ, dξ dξ dξ d σ df σ + df f γ A f + ξ. dξ dξ dξ A ad B are chose fro the syste of olear algebrac equatos γ B γ A σ + σ + /, /, ad /, /. The the doa Q accordg to the prcple of coparg solutos we have k ( η ) dη u t x u t x e f (, ) (, ) ( ), + k ( η ) dη u t x u t x e f (, ) (, ) ( ), + where t t f ( ξ ), f ( ξ ) ad τ ( t) -prevously defed fucto. Slow dffuso.case the soluto of the equato (4) obtaed the followg fuctos ξ ξ >, >, > (slow dffuso). Usg the ethod of olear splttg for θ ( ξ ) ( a ξ ), θ ( ξ ) ( a ξ ), + + where Ipact Factor (JCC): NAAS Ratg: 3.

5 Propertes of Self Slar Solutos of Reacto-Dffuso Systes of Quaslear Equatos 559 ( σ + ), ( σ + ), ( σ )( σ ) + +, ξ < a >, ( y) + ax ( y, ), a. It s kow that for the global exstece of a soluto of the task (4) of the (ξ fuctos f ) ust satsfy the followg equalty: d df ξ df µ dξ dξ dξ d df ξ df f + + µ f f f dξ dξ dξ σ ( f ) + + ( f f f ), σ ( ) ( ), ad where a >. /, /. It s show that the fuctos θ ( ξ ), θ ( ) Теорема. Fte soluto of the proble (4) at Fast dffuso.case ξ wll be asyptotc fte solutos. ξ a has asyptotcs ( )~ ( ) >, >, < (fast dffuso). For (4) there s χ ( ξ ) ( a + ξ ), χ ( ξ ) ( a + ξ ), ( σ + ) ( σ + ),, ( σ )( σ ) + +, f ξ θ ξ,,. Theore. At ξ + the vashg at fty soluto of the proble (4) has asyptotc ( )~ ( ) f ξ χ ξ. Ad also the doa Q{(t,: <t<, x R} cosdered parabolc syste of two quaslear equatos of the source ad absorpto task u t u t ( a x ( a x u u + a + a u u u ) x u ) x + ( b u + ( b u u + bu ) + k( t) u( u ) x u + bu ) + k ( t) u ( u ) x (5), a j, b j - postve real ubers,, u u t,, u t, ( u - desred soluto. At ( a, b or a, b atheatcal odel (5) s a syste of the type reacto-dffuso wth dffuso j j j j edtor@tjprc.org

6 56 D. K. Muhaedyeva coeffcets a u, b u. I the case where at least oe of the coeffcets a ad b (the sg ca j j be arbtrary), syste (5) s cross-dffuso (utual-dffuso for,j,). j j (4). Qualtatve propertes of the cosdered proble are studed by costructg self-slar syste of equatos for The, the obtaed self-slar syste d df df ξ df ( a f a f ) ( b f b f ) ψ f( f ), dξ dξ dξ + + dξ d df df ξ df ( a f + a f ) + ( b f + b f ) + + ψ f( f ), dξ dξ dξ dξ (6) adτ τ ( t) s chose as γ+ ( Τ + τ ), f γ +, γ+ τ( τ ) l( T + τ ), f γ +, ( T + τ ), f, γ γ. f Syste (6) has a approxate soluto of the for f A( a bξ ), f B( a bξ ) ( y) ax(, y). η η Case I η η, η, ad the coeffcets A ad B are detered fro the soluto of the systes of olear algebrac equatos [ aa + ab ] ηa + [ b A + b B ] [ a A + a B ] η A + [ b A + b B ] η B ( b) A, η B ( b) B. Case II a ; a ; b ; b ; a ; a ; b ; b ; η, η.. Ad the coeffcets A ad B are detered fro the soluto of the syste of olear algebrac equatos Ipact Factor (JCC): NAAS Ratg: 3.

7 Propertes of Self Slar Solutos of Reacto-Dffuso Systes of Quaslear Equatos 56 Case III a ( ) B η b, b ( ) A η b. a ; a ; b ; b ; a ; a ; b ; b ; + η (+ ), + η (+ ). Ad the coeffcets A ad B are detered fro the soluto of the syste of olear algebrac equatos b a B A + + η ( b) A, η ( b) B. Types of Self-Slar Reges wth Peakg There ca be three types of self-slar odes wth peakg: NS, S, ad LS. At < <,, HS ode s realzed. Studes have show that a self-slar proble ths case has a uque egefucto that decreases ootocally o a seget wth a axu at the ceter of syetry. A self-slar soluto s a wave whose apltude ad frot crease the blast ode. At,, takes place S ode. Self-slar soluto s a o-statoary dsspatve localzed structure. Isde the area of localzato, the uber of dvduals creases rege wth aggravato, ad beyod reas equal to zero. Self-slar soluto LS ode, s a o-statoary dsspatve structure; all pots of whch ove to the ceter τ of syetry, the soluto at T turs to fty a sgle pot the ceter of syetry. Self-slar solutos ca exst whe >,,. The followg are the results of uercal experets for dfferet values of the paraeters. To detere the uber of egefuctos (EF) self-slar proble LS ode, ad studes of the ature of ther depedece o the paraeter was coducted bfurcato aalyss of the solutos. The frst te egefuctos cotued o the paraeter for a fxed paraeter ( ). edtor@tjprc.org

8 56 D. K. Muhaedyeva Paraeter Values a ; a a ; a t ; b ; b ; b ; b a ; a a ; a t 4 ; b ; b ; b ; b t a ; a ; b ; b t 4 a ; a ; b ; b a ; a a ; a ; b ; b ; b ; b a ; a t 4 a ; a ; b ; b ; b ; b LS-rege, 3, k 3, k 9 eps 3 < < ˆ j ˆ + /( j ), j 3 j S-rege,, k, k 9 eps 3 S-rege 4, 4, k, k eps 3 HS-rege.7,.7.5, k.5, k 3 eps 3 < HS-rege.4,.4.5, k.7, k 5 eps 3 < Fgure : Results of Coputatoal Experet CONCLUSIONS The aalyss showed that the egefucto wth the uber j,3,4,... exsts the terval < ˆ <, where j Ipact Factor (JCC): NAAS Ratg: 3.

9 Propertes of Self Slar Solutos of Reacto-Dffuso Systes of Quaslear Equatos 563 ˆ + /( j ),,,,3,4,... j ˆ j (7) Values и j are the bfurcato pots whch the EF ceases to exst. The frst EF exsts for ay value >,,. It follows fro (7) that whe > ˆ have oly oe ege fucto.,, The greater the uber of SF, the arrower the terval of the paraeter whch t exsts. self-slar task LS ode ca The uber of egefuctos N that has a self-slar proble for the gve forula, where a ),, : /( ad s detered by the N [ a ], f a s ot a teger, a, f a s teger. The sequece of bfurcato pots that defe the rght boudary of the doa of exstece of egefuctos s a fte sequece that coverges to a pot + σ, whch s the coo left boudary of the tervals of exstece of all egefuctos the LS-ode. REFERENCES. Bareblatt G. I., Etov V. M., Ryzhk V. M. Theory of usteady fltrato of flud ad gas. M.: Nedra, 97.. Martso L. K. the Study of atheatcal odels of olear heat coducto eda wth voluetrc absorpto. I the book: Matheatcal odelg. Processes olear eda. M.: Nauka, 986, pp Olek O. A., Kalashkov A. S., Zhou Yu-L. The Cauchy proble ad boudary probles for equatos of the type of ostatoary fltrato. Izv. USSR ACADEMY OF SCIENCES. Ser. Mat. 958, vol., No. 5, pp Vazquez, Jua L., Galaktoov, V. A., Asyptotc behavor of solutos of olear equatos of the dffuso-absorpto at a crtcal dcator. //DAN SSSR, 99, vol. 34, o. 3, pp Galaktoov V. A. O soe propertes of travelg waves a edu wth olear theral coductvty ad heat sources. // JVM ad MF, 98, vol.. No. 4, pp Galaktoov, V. A., O codtos of o-exstece ad localzato of solutos of the Cauchy proble for a class of olear parabolc equatos. // JVM ad MF, 983, V. 3. No. 6, pp Alle W.C. Aal aggregato: a study geeral socology. Chcago: ChcagoUv.Press., Belotelov N. V., Locusts D. A. Lear stablty aalyss of systes wth dffuso o evroetal exaple. Bophyscs, 984, No., pp Murray George. Nolear dfferetal equatos bology. Lectures o odels. Moscow, Mr, 983, 394 p.. Arpov M. M. Method of stadard equatos for solutos of olear boudary value probles. Tashket: FAN, p.. 5.Arpov M, Sadullaeva Sh. Qualtatve propertes of solutos of a doubly olear reacto-dffuso syste wth a source. Joural of Appled Matheatcs ad Physcs, 3, 5, edtor@tjprc.org

10 564 D. K. Muhaedyeva. M. M. Arpov, Z. Rakhoov. About the asyptotcs of the solutos of heat coducto proble wth a source ad olear boudary codto. Coputg techology, Volue the Bullet of KazNU. Al-Farab, Seres atheatcs, echacs ad Iforatcs 3 (86), Part, 5, Arpov M., Matyakubov A. o the asyptotc behavor of olear parabolc equatos ot dvergece for. Coputg techology, Volue the Bullet of KazNU. Al-Farab, Seres atheatcs, echacs ad Iforatcs 3 (86), Part, 5, Arpov M., Mukhaedeva D.K. Populato Model of Kologorov-Fsher type wth Nolear Cross-dffuso. Matheatcs ad Coputers Scece ad Egeerg Seres, 4, 5, Arpov M.M., Muhaedyeva D.K. Populato odel of two copetg populatos wth double olear dffuso. Iteratoal scetfc ad techcal joural Checal techology. Cotrol ad aageet, 3-4 ad Joural of Korea ulteda socety South Korea, Seoul, 5, Arpov M.M., Mukhaedyeva D.K. Splttg algorth Kologorov-Fsher type reacto-dffuso task //Iteratoal Joural of Matheatcs ad Coputer Applcatos Research, Vol.3, 4, 3, р Arpov M.M., Mukhaedyeva D.K. To the uercal odelg of self-slar solutos of reacto-dffuso syste of the oe task of bologcal populato of Kologorov-Fsher type //Iteratoal Joural of Research Egeerg ad Techology, Vol.,, 3, p Arpov M.M, Muhaedeva D.K. Cross-dffuso odel wth double olearty hetrogeous evroet //Iteratoal scetfc-techcal joural of Checal techology. Motorg ad cotrol, No. 4, 4, pp Arpov M.M, Muhaedyeva D. K. Approaches to uercal odelg of the proble of reacto wth the dffuso of Kologorov-Fsher the two-desoal case //Uzbek joural probles of Iforatcs ad eergy, No. -, 4, pp Arpov M.M, Muhaedyeva D. K. Waves dffusve systes oe task of bologcal populato of Kologorov-Fsher dual olear cross-dffuso //Uzbek joural probles of Iforatcs ad eergy, No. 3-4, 5, pp. 44-5, (5..; No. 5).. Arpov M.M, Muhaedyeva D. K. Populato odel covectve trasfer of Kologorov-Fsher dual olear cross-dffuso //teratoal scetfc-techcal joural of Checal techology. Cotrol ad aageet, 5, 5, p. 6-, (5..; ).. Arpov M.M, Muhaedyeva D. K. Cross-dffuso odel of covectve trasfer wth double olearty. //Probles coputg ad appled atheatcs, No., 5, pp. 5-9, (5..; 3) 3. Kaptsa S. P., Kurdyuov S. P., Maletsk, G. G. Syergetcs ad forecasts of the future, M., S. P. Kurdyuov, E. S. Kurka, the Spectru of egefuctos of the self-slar proble for the olear equato of heat-coducto wth a source, Zh. Vychsl. od. ad od. Phys., 44:9 (4), ; Coput. Math. Phys., 44:9 (4), Modes wth peakg. The evoluto of the dea. Moscow: Nauka Dtreko N. V., Kurdyuov S. P., Mkhalov A. P., Saarsk A. A. Localzato of therouclear cobusto a plasa wth electroc theral coductvty //J. of Appl T. 6. Vol.9. C Tkhoov A. N., Saarsky A. A. zaklyaz'sky L. A. etc. o-lear effect of educato of self-sustag hgh-teperature electrcally coductg gas layer ostatoary processes of agetohydrodyacs // Doc. A SSSR, 967, Vol. 73, No. 4, Ipact Factor (JCC): NAAS Ratg: 3.

11 Propertes of Self Slar Solutos of Reacto-Dffuso Systes of Quaslear Equatos V. Volterra. Matheatcal theory of struggle for exstece-m.: Nauka, 976, 88 p. 9. Bazyk, A. D., Berezovskaya F. S. Allee Effect, lower crtcal populato sze ad dyacs of a syste a predator - vct. Probles of ecologcal otorg ad ecosyste odelg. - L.: Gdroeteozdat, 979, vol., pp Belotelov N. V., Lobaov A. I. Populato odel wth olear dffuso. // Matheatcal odelg. M., 997,, p edtor@tjprc.org

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