A fully discrete difference scheme for a diffusion-wave system

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1 Appled Nuercal Matheatcs 56 6) A fully dscrete dfference schee for a dffuson-wave syste Zh-zhong Sun a Xaonan Wu b a Departent of Matheatcs Southeast Unversty Nanjng 96 PR Chna b Departent of Matheatcs Hong Kong Baptst Unversty Kowloon Tong Hong Kong Avalable onlne 9 Aprl 5 Abstract A fully dscrete dfference schee s derved for a dffuson-wave syste by ntroducng two new varables to transfor the orgnal equaton nto a low order syste of equatons. The solvablty stablty and L convergence are proved by the energy ethod. Slar results are provded for a slow dffuson syste. A nuercal exaple deonstrates the theoretcal results. 5 IMACS. Publshed by Elsever B.V. All rghts reserved. Keywords: Dffuson-wave syste; Fnte dfference; Convergence; Solvablty; Stablty. Introducton Ths artcle s concerned wth a nuercal soluton to a fractonal dffuson-wave FDW) syste subjected to a non-hoogeneous feld. A fractonal dffuson-wave equaton s a lnear ntegro-partal dfferental equaton obtaned fro the classcal dffuson or wave equaton by replacng the frst- or second-order te dervatve by a fractonal dervatve of order α> 3. Oldha and Spaner consdered a fractonal dffuson equaton that contans frst order dervatve n space and half order dervatve n te. Ngatulln 89 ponted out that any of the unversal electroagnetc acoustc and * Correspondng author. E-al addresses: zzsun@seu.edu.cn Z.-Z. Sun) xwu@hkbu.edu.hk X. Wu). Research s supported Natonal Natural Scence Foundaton of Chna Contract grant nuber 473) and by Research Foundaton of Southeast Unversty Contract grant nuber XJ373) and RGC of Hong Kong. Research s supported by RGC of Hong Kong and FRG of Hong Kong Baptst Unversty /$3. 5 IMACS. Publshed by Elsever B.V. All rghts reserved. do:.6/j.apnu.5.3.3

2 94 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 echancal responses can be odelled accurately usng the FDW equatons. Wess 4 and Schneder and Wess presented solutons of FDW equatons n ters of Fox s H -functons. Fujta 5 presented the exstence and the unqueness of the soluton of the Cauchy proble of the followng type: α ux t)/ t α β ux t)/ x β α β. The results presented offer an nterpretaton to phenoena between the heat equaton α β ) and the wave equaton α β ). Fujta 67 consdered ntegro-dfferental equatons whch exhbt heat dffuson and wave propagaton propertes. He also deonstrated that certan operators assocated wth the equatons can be decoposed and the solutons can be wrtten as the su of the solutons of the decoposed operators. Gnoa et al. 8 presented a fractonal dffuson equaton descrbng relaxaton phenoena n vscoelastc aterals. Mbodje and Montseny 6 nvestgated the exstence unqueness and asyptotc decay of the wave equaton wth fractonal dervatve feedback and showed that the ethod developed can be easly adapted to a wde class of probles nvolvng fractonal dervatve or ntegral operators of the te varable. Manard used Laplace transfor ethod to obtan the fundaental soluton of the FDW equatons and expressed the n ters of auxlary functon Mzβ) where z x /t β s the slarty varable. He further showed that such a functon s an entre functon of Wrght type. Agrawal presented a general soluton to FDW equatons contanng fourth order space dervatve defned n unbounded and bounded doans. Metzler and Klafter 7 used Fourer Laplace transfor and the separaton of varables to solve the fractonal dffuson equaton for absorbng and reflectng boundary value probles. Helfer 9 presented the soluton of a fractonal dffuson proble n ters of H -functons. Manard et al. 4 presented the fundaental soluton Green functon) for the space te fractonal dffuson equaton whch s obtaned fro the standard dffuson equaton by replacng the second-order space dervatve wth a Resz Feller dervatve of order α and skewness θ θ nα α}) and the frst-order te dervatve wth a fractonal dervatve of order β. They also presented explct forulae for varous functons n ters of paraeters α θ and β. Agrawal 3 used the ethod of separaton of varables to dentfy the egenfunctons and to reduce the dfferental equaton of an FDW nto a set of nfnte equatons each of whch descrbes the dynacs of an egenfuncton. A Laplace transfor technque s used to obtan the fractonal Green s functon and a Duhael ntegral type expresson for the syste s response. Copared wth the consderable work on the theoretcal analyss only a lttle work has been done on the nuercal ethod. Sanz-Serna presented a teporal se-dscrete algorth and proved the one order convergence. The lnear equaton nvestgated by h could be consdered as a 3/ order tefractonal dffuson-wave equaton. Sae probles are nvestgated by Lopez-Marcos and Tang 3. A backward-euler schee and a Crank Ncolson schee are presented n and 3 respectvely. The stablty and convergence are obtaned. Bechelova 4 proposed a dfference schee for the xed boundary value proble of an α <α<) order te fractonal dffuson equaton a slow dffuson syste) and proved the O α + h ) order condtonal convergence n the unfor etrc by the axu prncple. In ths artcle we gve a fully dscrete dfference schee for the FDW equaton and prove that the dfference schee s unquely solvable uncondtonally stable and convergent n L nor. The convergence order s O 3α + h ).

3 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Consder the FDW equaton 3 α u c t u α x + fxt) K x L t >.) along wth the ntal condtons ux) ux ) φx) ψx) x L.) t and the boundary condtons ut) ult) t >.3) where c and K are constants of densons Length Te α and Length x L and t> are space and te varables φ) φl) u ux t) and fxt)are the feld varables and α u t t α Ɣ α) ux s) s ds t s) α <α< wth Ɣ denotng the gaa functon. When α Eq..) represents a dffuson equaton c and fxt) are called the dffuson coeffcent and the source ter respectvely. When α Eq..) represents a wave equaton c and fxt) denote the square of the wave velocty and an external force feld respectvely. For <α< the fractonal equaton n.) s expected to nterpolate the dffuson equaton and the wave equaton thus n ths case t could be referred to as the te-fractonal dffusonwave equaton. We have to pont out that for <α< both the ntal condtons n.) are necessary as the wave equaton α ) but for <α< only ntal condton ux ) φx) can be posed as the dffuson equaton α ) see e.g. 53). Let ω h x M} s a unfor esh of the nterval L where x h M wth h L/M. Letω t n n }wheret n n >. Suppose u u n M n } s a grd functon on ω h ω. Introduce the followng notatons: u n/ u n + u n ) δt u n/ u n u n ) δ x u n / h ) u n u n δ x u n δx u n +/ h δ ) xu n / where u n/ s an average of u at the ponts x t n ) and x t n ) and δ t u n/ s the dfference quotent of u based on these two ponts; δ x u n / s the frst-order dfference quotent of u on the ponts x t n ) and x t n ) and δx un s the second-order dfference quotent at the ponts x t n ) x t n ) and x + t n ). We also defne u n ax M u n In addton f u n andun M we have u n L δx u n. h δx u n M ). δx u n /.4)

4 96 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 The dfference schee we wll consder for.).3) s as follows: a δ t u n/ a nk a nk )δ t u k/ a n ψ cɣ α) M n u φ M u n un M n where and and a l t l+ t l dt t α α δ x un/ + K f n/.5).6).7) tl+ ) α t l ) α α l + ) α l α l.8) α φ φx ) M; ψ ψx ) f n/ It s easy to know that f x t ) n + t n M n. a α α a l >a l+ l.9) a nk a nk ) a a n n..) At each te level.5).7) s a trdagonal syste of lnear algebrac equatons whch can be solved by the double sweep ethod Thoas algorth). In the followng for splcty we suppose that proble.).3) has soluton ux t) Cxt 43 L )). The reander of the artcle s arranged as follows. In Secton the dfference schee.5).7) s derved by ntroducng two new varables and transforng the orgnal equaton.) nto a low order syste of equatons. In Secton 3 the an results are proved whch are Theores 3. and 3.3. In Secton 4 slar results are presented for a slow dffuson syste. Secton 5 provdes a nuercal exaple to deonstrate the theoretcal results.. The dervaton of the dfference schee For the dervaton of the dfference schee we need the followng leas. Lea.. For n and t k k k n we have t k t t n t) α tk α + 3α 3 α + α) t n t k ) α + t } k t t n t k ) α dt 3α.

5 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Proof. Let gt) t n t) α then t tk gt) gt k ) + t k t gt k ) g ξ k )t t k )t t k ) α)α )t n ξ k ) α t k t)t t k ) where ξ k t k t k ) t t k t k ). Fro the above nequalty we have n t k n t tk gt) t k gt k ) + t } k t gt k ) dt α)α )t n ξ k ) α t k t)t t k ) dt n α)α ) t n t k ) α Snce t k t k t)t t k ) dt 3 /6 we obtan n t k t tk gt) t k t k t)t t k ) dt. gt k ) + t } k t gt k ) dt α)α ) n 3 t n t k ) α α)α ) knt k t n t n t t n t) α dt α) t n t n ) α t n t ) α α 3α..) On the other hand t tk gt) gt k ) + t } k t gt k ) dt t n t n gt)dt gt n) + gt n ) t n t) α dt t n t n ) α + t n t n ) α t n 3α 3 α + α) 3α..)

6 98 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 The lea follows fro.) and.): t k n t tk gt) + kn ) tk t k gt k ) + t } k t gt k ) dt t tk gt) α + 3α 3 α + α) 3α. Lea.. Suppose gt) C t n. Then t n g dt t) t n t) gt k ) gt k ) α α α + 3α 3 α + α) Proof. For splcty denote A t n gt k ) + t } k t gt k ) dt t k g dt t) t n t) gt k ) gt k ) α dt t n t) α ax t t n g t) 3α. Usng Taylor expanson wth ntegral reander we have g t) gt k) gt k ) t g s)s t k ) ds whch yelds A t k t k t k g t) gt k) gt k ) t t k g s)s t k ) ds Exchangng the order of ntegraton we get A α t k t k dt t n t) α s t n s) α tk t dt t n t) α. t g s)t k s)ds g dt s)t k s)ds t n t). α t n t k ) α + t k s t n t k ) }g α s) ds.

7 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Applyng Lea. we obtan the result: A α t k α ax t t t n g t) s t n s) α tk t k α α + 3α 3 α + α) t n t k ) α + t } k s g t n t k ) α s) ds s t n s) α tk ax g t) 3α. t t n Lea.3. Suppose gt) C t n. Then t n g dt t) t n t) a α gt n ) a nk a nk )gt k ) a n gt ) α α + 3α 3 α + α) where a l s defned n.8) and <α<. ax g t) 3α t t n t n t k ) α + t } k s t n t k ) α ds Proof. Observng Lea. t suffces to verfy gt k ) gt k ) t k dt t n t) a α gt n ) a nk a nk )gt k ) a n gt ). In fact t k gt k ) gt k ) t k dt t n t) α gt k ) gt k ) tnk+ ) α t nk ) α α a gt n ) a nk a nk )gt k ) a n gt ). Ths copletes the proof. a nk gtk ) gt k ) ) Let vxt) uxt) t.3)

8 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 and wxt) Then.) becoes t Ɣ α) vxs) s ds t s) α..4) c wxt) ux t) + fxt)..5) x K Defne the grd functons U n ux t n ) V n vx t n ) W n wx t n ) M n. Usng Taylor expanson t follows fro.3) and.5) that V n/ δ t U n/ + r ) n/ and c W n/ δx U n/ + K f n/ + r ) n/.7) and there exsts a constant c such that r ) n/ c.8) and r ) n/ c + h )..9) Based on Lea.3 we have W n Consequently t n Ɣ α) vx t) t a V n Ɣ α) W n/ W n + W n ) Ɣ α) dt t n t) α a V n/ and there exsts a constant c such that r3 ) n/ c 3α. a nk a nk )V k a n V + O 3α) n. a nk a nk )V k/ a n V.6) + r 3 ) n/.).)

9 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 Substtutng.6) nto.) we have W n/ a δ t U n/ Ɣ α) + a r ) n/ Ɣ α) a nk a nk )δ t U k/ a n V a nk a nk )r ) k/ Then substtutng above result nto.7) and notcng V vx ) ψx ) we obtan a δ t U n/ a nk a nk )δ t U k/ a n ψx ) cɣ α) where δ x U n/ R n/ + K f n/ a r ) n/ c Ɣ α) + r 3 ) n/..) + R n/ M n.3) a nk a nk )r ) k/ + r 3 ) n/ } + r ) n/. Accordng.9).).8).9) and.) we have n/ R c h c α)ɣ α) + c + cc + 3α)..4) In addton fro.).3) we have U φx ) M.5) and U n Un M n..6) Observng.3) and.5).6) t s natural to construct the dfference schee.5).7) for the proble.).3). 3. Analyss of the dfference schee Before we prove the solvablty stablty and convergence we gve the followng leas. Lea 3.. For any G G G G 3...} and q we have a G n a nk a nk )G k a n q G n t α N N G n t α N α) q N 3... where a l s defned n.8).

10 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 Proof. a G n a nk a nk )G k a n q G n N a G n a nk a nk )G k G n n a G n a G n a n qg n a nk a nk ) G k + ) G n n a nk a nk )G n n a n q a n G n. ) a n q + G n a nk a nk )G k Exchangng the suaton order of the thrd ter n the last nequalty we obtan a G n a nk a nk )G k a n q a G n G n n a a n )G n N a a Nk )G k n a Nn G n t α N N G n a n q a N t α N α) q G n t α N α) q wherewehaveusedthata l } s a strctly decreasng sequence and N a n n t N a n t dξ ξ α tn α α a l t l+ t l a n q x α dx t α l+. a n G n Lea 3.. Suppose u n } s the soluton of a δ t u n/ a nk a nk )δ t u k/ a n q δx cɣ α) un/ + P n/ M n... 3.) u φ M 3.) u n un M k )

11 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) We have δ x u n δ x u t α n + c α)ɣ α) h q + cɣ α)tn α n. k/) h P 3.4) Proof. Multplyng 3.) by hδ t u n/ and sung up for fro to M and for n fro to we obtan cɣ α) h h δt u n/ a δ t u n/ ) δ x u n/ } a nk a nk )δ t u k/ a n q δ t u n/ ) + h δt u n/ ) n/ P. 3.5) Usng Lea 3. we have } h a δ t u n/ a nk a nk )δ t u k/ a n q δ t u n/ } h t α δt u n/ ) t α α) q t α δt u n/ t α α) h q. 3.6) Applyng 3.) and 3.3) we have δ t u n/ andδ t u n/ M. Consequently h δt u n/ ) δ x u n/ M h δx u n/ / ) δt δx u n/ ) / M h δx u/) n M h δx u n / ) δ x u δ x u ). 3.7) In addton h h δt u n/ ) P n/ cɣ α) t α δt u n/ ) + cɣ α)tα cɣ α) t α δt u n/ + cɣ α)tα n/) P h P n/ ). 3.8)

12 4 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 Substtutng 3.6) 3.8) nto 3.5) we obtan cɣ α) t α δt u n/ or + c δ x u δ x u ) + t α Ɣ α)tα δx u n δx u + n. Ths copletes the proof. t α α) h q } cɣ α) δ t u n/ h P n/ tn α c α)ɣ α) h ) q + cɣ α)tn α Theore 3.. The dfference schee.5).7) s unquely solvable. k/) h P Proof. Snce.5).7) s a syste of lnear algebrac equatons at each te level t suffces to show that the correspondng hoogeneous equatons: a δ t u n/ a nk a nk )δ t u k/ δx cɣ α) un/ M n 3.9) u M 3.) u n un M n 3.) have only zero soluton. Usng Lea 3. we have δx u n n 3... Cobnng the above equalty wth 3.) arrves at u n M n. Theore 3.. Let u n M n } be the soluton of the dfference schee.5).7). Then we have δx u n δx u tn α + c α)ɣ α) h ψ cɣ α) + t α k/) K n h f n. Proof. The result needed can be easly obtaned fro Lea 3..

13 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Theore 3.3. Let.).3) have soluton ux t) Cxt 43 L T) and u n M n } be the soluton of the dfference schee.5).7). Then for n T we have ax ux t n ) u n L M Proof. Denote c α)ɣ α) + c + cc T α Ɣ α) h + 3α). c ũ n U n u n M n. Subtractng.5).7) fro.3).5).6) respectvely we obtan the error equatons a δ t ũ n/ cɣ α) M n ũ M a nk a nk )δ t ũ k/ ũ n ũn M n. Usng Lea 3. we have δ x ũ n cɣ α)tn α k/) h R n. δ xũn/ + R n/ Insertng.4) nto the rght hand of the above nequalty we get δx ũ n c Lt α)ɣ α) + c α + cc n Ɣ α) h + 3α) c n. Notcng.4) we have the result: ũn L c T α α)ɣ α) + c Ɣ α) + cc h + 3α) c when n T. 4. A slow dffuson syste Consder the slow dffuson equaton 34 α u c t u α x + fxt) K x L t > 4.) along wth the ntal condton ux ) φx) and the boundary condtons x L ut) ult) t > 4.3) 4.)

14 6 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 where α u t α Ɣ α) t uxs) s ds t s) α <α<. Let α be replaced by α + n Lea.3 and n Lea 3. we have Lea 4.. Suppose gt) C t n. Then t n g dt t) t n t) b α gt n ) b nk b nk )gt k ) b n gt ) α α + α α + α) ax g t) α t t n where <α< and b l t l+ t l dt t α α tl+ ) α t l ) α α l + ) α l α l. 4.4) α Lea 4.. For any G G G G 3...} and q we have b G n b nk b nk )G k b n q G n t α N N G n t α N α) q N 3... where <α< and b l s defned n 4.4). Usng Lea 4. and slarly to the dervaton of the dfference schee.5).7) for the proble.).3) n Secton we ay construct the followng dfference schee for 4.) 4.3): n b u n cɣ α) b nk b nk )u k b nu δx un + K f n M n 4.5) u φ M 4.6) u n un M n. 4.7) Multplyng 4.5) by hu n usng Lea 4. and slarly to the analyss n Secton 3 we can prove the followng theore: Theore 4.. The dfference schee 4.5) 4.7) s unquely solvable and the followng estate s vald: δx u k t α n c α)ɣ α) h φ + cɣ α)tα ) n h f k K n. 4.8)

15 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Furtherore f 4.) 4.3) has soluton ux t) Cxt 4 L T) then the soluton of the dfference schee 4.5) 4.7) converges to the soluton of the proble 4.) 4.3) wth the convergence order of Oh + α ) n L -nor n the sense that U u) k C h + α) when n T where C s a constant ndependent of h and. 5. Nuercal exaple In order to deonstrate the effectveness of our dfference schee we copute the followng proble: α u t u + snπx) x <t 5.) α x ux) ux ) x 5.) t ut) ut) t. 5.3) The exact soluton of the above proble s 3 where ux t) π Eα π t α) snπx) 5.4) E α z) k z k Ɣαk + ). If α 3/ then 5.4) can be expressed as follows: ux t) π t α ) π π 3 ) π t α ) 3 snπx). 5.5) Table Soe nuercal results M \ x t) 8 ) 8 ) 3 8 ) 4 8 ) D.8566D.486D+.35D D.8D.4554D+.368D D D.445D+.358D D.79974D.447D+.3D D.79984D.445D+.38D D.79954D.44D+.33D+ Exact soluton.43436d.79937d.4399d+.3d+

16 8 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) 93 9 Fg.. The curves of the errors of the fnte solutons at t. Table The axu errors u u h M u u h D D D D D D5 Take α 3/ h /M. Table presents the nuercal and exact solutons at soe ponts for dfferent esh szes. Fg. plots the curves of the errors of the dfference solutons on the lne t for dfferent esh szes. Table gves the axal errors of dfference solutons at all esh ponts for dfferent esh szes. In Table the axal error s defned as follows u u h ax ax ux t n ) u n }. n M M It s clear that the fnte dfference soluton s very accurate and converges quckly to the exact soluton. Suppose u u h ch p. Then we have log u u h log c + p log h). Usng the data n Table and wth the help of MATLAB we obtan lnear fttng functons log u u h log h).

17 Z.-Z. Sun X. Wu / Appled Nuercal Matheatcs 56 6) Concluson In ths artcle we present a dfference schee for the ntal-boundary value proble of a dffusonwave equaton. The solvablty stablty and convergence are proved by the energy ethod. Slar results are gven for a slow dffuson syste. A nuercal exaple deonstrates the theoretcal results. References O.P. Agrawal A general soluton for the fourth-order fractonal dffuson-wave equaton Fract. Calculus Appl. Anal. 3 ) ). O.P. Agrawal A general soluton for the fourth-order fractonal dffuson-wave equaton defned n bounded doan Coput. Struct. 79 ) O.P. Agrawal Response of a dffuson-wave syste subjected to deternstc and stochastc felds Z. Angew. Math. Mech. 83 4) 3) A.R. Bechelova On the convergence of dfference schees for the dffuson equaton for fractonal order Ukranan Math. J. 5 7) 998) Y. Fujta Cauchy probles of fractonal order and stable processes Japan J. Appl. Math. 7 3) 99) Y. Fujta Integrodfferental equaton whch nterpolates the heat equaton and the wave equaton Osaka J. Math. 7 ) 99) Y. Fujta Integrodfferental equaton whch nterpolates the heat equaton and the wave equaton II Osaka J. Math. 7 4) ) M. Gnoa S. Cerbell H.E. Roan Fractonal dffuson equaton and relaxaton n coplex vscoelastc aterals Physca A 999) R. Helfer Fractonal dffuson based on Reann Louvlle fractonal dervatves J. Phys. Che. 4 ) J.C. Lopez-Marcos A dfference schee for a nonlnear partal ntegrodfferental equaton SIAM J. Nuer. Anal. 7 ) 99) 3. F. Manard Fractonal relaxaton-oscllaton and fractonal dffuson-wave phenoena Chaos Soltons Fractals 7 9) 996) F. Manard The fundaental solutons for the fractonal dffuson-wave equaton Appl. Math. Lett. 9 6) 996) F. Manard Soe basc probles n contnuu and statstcal echancs n: A. Carpnter F. Manard Eds.) Fractals and Fractonal Calculus n Contnuu Mechancs Sprnger Wen 997 pp F. Manard Y. Luchko G. Pagnn The fundaental soluton of the space-te fractonal dffuson equaton Fract. Calculus Appl. Anal. 4 ) F. Manard G. Pagnn The wrght functons as solutons of the te-fractonal dffuson equaton Appl. Math. Coput. 4 3) B. Mbodje G. Montseny Boundary fractonal dervatve control of the wave equaton IEEE Trans. Autoatc Control 4 995) R. Metzler J. Klafter Boundary value probles for fractonal dffuson equatons Physca A 78 ) R.R. Ngatulln To the theoretcal explanaton of the unversal response Physca Status B): Basc Res. 3 ) 984) R.R. Ngatulln Realzaton of the generalzed transfer equaton n a edu wth fractal geoetry Physca Status B): Basc Res. 33 ) 986) K.B. Oldha J. Spaner The Fractonal Calculus Acadec Press New York 999. J.M. Sanz-Serna A nuercal ethod for a partal ntegro-dfferental equatons SIAM J. Nuer. Anal ) W.R. Schneder W. Wess Fractonal dffuson and wave equatons J. Math. Phys. 3 ) 989) T. Tang A fnte dfference schee for partal ntegro-dfferental equatons wth a weakly sngular kernel Appl. Nuer. Math. 993) W. Wess The fractonal dffuson equaton J. Math. Phys. 7 ) 996)