Fractional Order Finite Difference Scheme For Soil Moisture Diffusion Equation And Its Applications

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1 IOS Joural of Mathematcs (IOS-JM e-iss: Volume 5, Issue 4 (Ja. - Feb. 3, PP -8 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its Applcatos S.M.Jogdad, K.C.Takale, V.C.Borkar 3. Departmet of Mathematcs, S. S. G. M. College, oha, aded (Maharashtra, Ida.. Departmet of Mathematcs, C Arts, JDB Commerce ad SC Scece College ashk-oad, (Maharashtra,Ida. 3. Departmet of Mathematcs, Yeshwat Mahavdyalya, aded(maharashtra, Ida. Abstract: The am of ths paper s to develop the mplct fte dfferece scheme for space fractoal sol mosture dffuso equato (SFSMD wth tal ad boudary codtos. We prove that the scheme s ucodtoally stable ad coverget. Also, as a applcato of ths scheme umercal soluto for space fractoal sol mosture dffuso equato s obtaed by Mathematca software. Keywords: Space fractoal, Sol mosture dffuso equato, Fte dfferece scheme, Fractoal dervatves, Mathematca. I. Itroducto Fractoal order partal dfferetal equatos have recetly foud ew applcatos hydrology, scece, egeerg ad face [, 5, 7, 8, 9]. A physcal/mathematcal approach to aomalous dffuso s based o geeralzed dffuso equato cotag dervatves of fractoal order space or tme or space-tme. ecetly fractoal dffuso equato have bee studed by may authors ad developed a fractoal order fte dfferece schemes for fractoal dffuso equatos. Therefore, ths coecto we develop space-fractoal mplct fte dfferece scheme for sol mosture dffuso equato of fractoal order. The soluto of the lear partal dfferetal equato of flow was frst proposed by Casagrade through the use of the graphcal flowet method [4]. Ths method s based o the assumptos that water flows rego must be defed terms of head or o-head flow. The flowet solutos proposed by Casagrade were for smple ucofed flow cases wthout flu boudary codtos. Frst epermetal study o the movemet of water the sol was doe by Hery Darcy (856. dgar Buckghm (97 descrbed the water flow usaturated porous meda modfyg the equato of Darcy. chard s (93 combed the equatos of Darcy ad Buckgham wth the equato of cotuty to establsh a overall relatoshp. Klute (97 descrbed several methods for estmatg the hydraulc coductvty ad dffusvty for usaturated sols [, 3, 4, 6]. To uderstad such pheomeo, sol scetsts have made some models for the flow of water to sol. Furthermore, may researchers developed dfferet types of equatos that models the water flow to sol. We cosder the geeral dffuso equato of usaturated flow of sol mosture as follows D D y y K D z z z t where, (, y, z, = the volumetrc sol mosture cotet, D = the dffusvty of sol mosture, D = D( s a fucto of mosture cotet ad K = the capllary or hydraulc coductvty of sol mosture. If for equato (., the flow takes place the Z drecto, as for fltrato of water to the sol, the the equato (. becomes oe-dmesoal flow equato, whch s gve below (. where K D z z z t h D K t h t = the teso head ad (. Page

2 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its K = the capllary coductvty. If the flow s cosdered drecto (take horzotal the equato (. becomes D t ow we assume that D s a costat the the oe-dmesoal dffuso equato s D t whch s eactly the dffuso heat flow equato [4] ad t s well studed by chard s [] for water flow stead of heat flow. The model problem for the mosture flow horzotal tube s gve by D, t, t We solve the partcular model problem of mosture flow to a horzotal tube, we eed to mpose proper tal ad boudary codtos. For that wth a tal uform mosture percetage of θ s ( s costa ad for whch at tme t =, become tal codto ad whch s mathematcally epressed as follows (.3 (.5, t, t, (.6 For left boudary codto, there s appled a source of water placed at = so as to mata at all tmes after t = s, ad whch s mathematcally epressed as follows, t,, t (.7 For the rght boudary codto, there s appled a source of water placed at sem fte plae so as to mata at all tmes after t = s, ths s mathematcally epressed as follows, t,, t (.8 Therefore, we have the model IBVP for sol mosture flow whch s gve as follows Subect to the tal ad boudary codtos D, t, t (.4 (.9, t, t, (., t,, t,, t,, t (. Where θ(, s volumetrc water cotet ad D s the dffusvty costat of sol mosture. I the et secto, we develop the fractoal order mplct fte dfferece scheme (SFIFDS for space fractoal sol mosture dffuso equato. The pla of the paper s as follows: I secto, the fractoal order mplct fte dfferece scheme s develop for space fractoal sol mosture dffuso equato. The secto 3, s devoted for stablty of the scheme ad the questo of covergece s proved secto 4. umercal soluto of space fractoal sol mosture dffuso equato s obtaed usg Mathematca software the last secto. II. Fte Dfferece Scheme We cosder the space fractoal sol mosture dffuso equato (SFSMD wth tal ad boudary codtos as follows (, (, D, t,, t tal codto: (,, (. boudary codtos: (,, : (,,, t (.3 where D s the dffusvty costat. We descretse the spatal α-order fractoal dervatve usg the Grüwald fte dfferece formula at all tme levels. The stadard Grüwald estmate geerally yelds ustable fte dfferece equato regardless of whatever result fte dfferece method s a eplct or a mplct system for related dscusso []. Therefore we use a rght shfted Grüwald formula to estmate the spatal α-order fractoal dervatve (. 3 Page

3 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its (, ( k lm ( ( k h, ( h k ( k ( where s the postve teger, h ad Γ(. s the gamma fucto. For the mplct umercal appromato scheme, we defe t be the tegrato tme t T ad Δ = h > to be the grd sze -drecto, ( - h wth h for =,,...,. Defe (, t ad let, t ( deote the umercal appromato to the eact soluto. We also defe the ormalzed Grüwald weghts by ( k g, k,k =,,... ( ( k For D (, tk, we adopt the rght shfted Grüwald formula at all tme levels for appromatg the secod order space dervatve by mplct type umercal appromato to equato (., we get D, where the above fractoal partal dfferetal operator s defed as, g, k k h k whch s a O ( h appromato to the α-order fractoal dervatve. Therefore, the fractoal appromated equato s D = g, k k h k After smplfcato, we get r g, k k D where r. h The tal codto s appromated as k,,...,,,,,... =, =,,...,. The left boudary codto s appromated as =, =,,,...,. ow usg cetral dfferece the rght boudary codto s appromated as follows,,,,..., h Therefore, the fractoal appromated IBVP s r g, k k k,,...,,,,,... tal codto: =, =,,...,. (.5 boudary codtos: = ad = (.6 D where r. h Therefore, the fractoal appromated IBVP (.4 (.6 ca be wrtte the followg matr equato form A (.4 ( Page

4 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its T where (,,,..., ad A ( a s a square matr of coeffcet of order. For =,,,...,, =,,,..., the coeffcets are, whe rg, whe a (.8 rg, whe,,4,... rgk, otherwse k,3,4,...,. Whle a, g g The above system of algebrac equatos s solved by usg Mathematca software secto 5. III. Stablty Ths secto s devoted for the stablty of the fractoal mplct fte dfferece scheme (.4 (.6 for the space fractoal sol mosture dffuso equato (SFSMD (.9 (.. emma 3.: If ( A,,,..., represets ege value of matr A the we prove the followg results: ( ( A,,,..., ( A Proof: The Gerschgor theorem states that each egevalue λ of a square matr A s at least oe of the followg dsk a al,,,..., M (3. l, l Therefore, each egevalue λ of matr A satsfes at least oe of the followg equaltes: a al al l, l l (3. a a a l, l a l (3.3 To prove (, we use equato (3.3 to matr A, the each egevalue λ of matr A satsfes the followg equalty. ( rg rg A rg rg r ( g g,(sce g g ( A... ( rg rg rg A g ( rg rg r ( g g g,(sce g g g ( A rg ( A rg rg rg... g g g g... g g 5 Page

5 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its Therefore, ths proves ( A,,,..., To prove (, we have A ma ( J A A ( A Theorem 3. The soluto of the fractoal appromated IBVP (.4 (.6 s ucodtoally stable. Proof: To prove that the above scheme s ucodtoally stable. We must show that for From the equato (.7, we have A,,,... Clearly, matr A s vertble. ow for =,,... from equato (3.4, we get... A A A A A ( A ( A ( A, (3.4 (3.5 From equato (3.5, we get A, ( By lemma 3., A Ths shows that the fte dfferece scheme for fractoal equato s ucodtoally stable. Hece the proof s completed. IV. Covergece I ths secto we dscuss the covergece of the fte dfferece scheme. Cosder the aother vector (, t t t T,..., (,,..., (, whch represets the eact soluto at tme level t, whose sze s. The fte dfferece scheme (3.4 wll become A,,,... (4. where s the vector of the trucato errors at level t. Theorem 4. The fractoal order fte dfferece scheme (.4 (.6 for SFSMD s coverget. Proof: If we subtract (3.4 from (4., we get ( A ( (4. Cosder the error vector, from equato (4., we get A ( Page

6 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its I equato (4.3, puttg =,,, we get A A A A A ( A A A [ A A ] A A [ A ] We take, the Sce by emma (3., Therefore, from equato (4.5, we get The proof s completed. ( A A ( A k k k s a zero vector, the from (4.4, we get k A A. ma k M A ad lm, ( M ( h, (, M as ( h, (, M (4.4 (4.5 V. umercal Solutos I ths secto, we obta the appromated soluto of space fractoal sol mosture dffuso equato wth tal ad boudary codtos. To obta the umercal soluto of the space fractoal sol mosture dffuso equato (SFSMD by the fte dfferece scheme, t s mportat to use some aalytcal model. Therefore, we preset a eample to demostrate that SFSMD ca be appled to smulate behavor of a fractoal dffuso equato by usg Mathematca Software. We cosder the followg, dmesoless oedmesoal space fractoal sol mosture dffuso equato wth sutable tal ad boudary boudary codtos (, (, D,,, t t tal codto : (,, boudary codtos : (,, (,, as, t wth the dffuso coeffcet D =. The umercal soluto obtaed at t =.5 by cosderg the parameters τ =.5, h =., α =.7,.8 ad.9, whch s smulated the followg fgure. 7 Page

7 Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its Fg.5. : The sol mosture dffuso profle wth t =.5, h =., α =.7(blue α =.8(red ad α =.9(gree VI. Coclusos. ( We develop fractoal order fte dfferece scheme for space fractoal sol mosture dffuso equato. ( The umercal eample s aalyzed to show that the umercal results are good agreemet wth theoretcal aalyss. ( The fractoal order mplct fte dfferece scheme s umercally stable. efereces. [] Florca MAT ad BDI, Dfferetal equatos ad ther applcatos to the SolMosture Study, Bullet ASVM, Hortculture 65((8. []. Hlfer, Applcatos of Fractoal Calculus Physcs, World Scetfc, Sgapore (. [3] Dael Hllel, Itroducto to Sol Physcs, Academc Press (98. [4] Do Krkham ad W.. Powers, Advaced Sol Physcs, Wley-Iterscece (97. [5] F.u, P. Zhuag, V. Ah, I, Turer, A Fractoal Order Implct Dfferece Appromato for the Space-Tme Fractoal Dffuso equato, AZIAM J.47 (MAC5, pp. C48-C68:(6. [6] Pater A.C., aats ad Martus TH. Ve Geuchte, Mlestoes Sol Physcs, J. Sol Scece 7,(6. [7] I. Podluby, Fractoal Dfferetal equatos, Academc Press, Sa Dago (999. [8] B.oss (d, Fractoal Calculus ad Its Applcatos, ecture otes Mathematcs, Vol.457, Sprger-Verlag, ew York ( 975. [9] S. She, F.u, rror Aalyss of a eplct Fte Dfferece Appromato for the SpaceFractoal Dffuso equato wth sulated eds, AZIAM J.46 (, pp. C87-C887:( Page