A numerical method for space fractional diffusion equations using a semidisrete scheme and Chebyshev collocation method

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1 Joual of mahemac ad comue cece A umecal mehod fo ace facoal dffuo equao ug a emee cheme ad Chebyhe collocao mehod Had Azz Ghaem Bad Loghma Deame of Mahemac af Bach Ilamc Azad Uey af Ia Hadazz36@yahoocom Deame of Mahemac Yazd Uey Yazd Ia Loghma@yazduac Acle hoy: Receed Jue 3 Acceed July 3 Aalable ole July 3 Abac I he ee ae a umecal aoach o effcely calculae he oluo of ace facoal dffuo equao egaed he fe dffeece cheme ad Chebyhe collocao mehod aled o ole h oblem Alo he ma fom of he ooed mehod obaed he umecal eamle ad comao wh ohe meho how ha he ee mehod effece Keywo: Facoal dffuo equao Fe dffeece Collocao Chebyhe olyomal Ioduco he ue of facoal aal dffeeal equao FPDE mahemac hyc egeeg ad chemy ha become ceagly oula ece yea o oba a aalycal oluo of h oblem eemely dffcul hu may auho ae eeg way o umecally ole hee oblem Some dffee umecal meho hae bee ooed fo olg FPDE Lu ad e al ooed mehod of le o afom he ace facoal Foe-Plac equao o a yem of oday dffeeal equao 7 F ad Roo 5 deeloed a lea quae fe eleme oluo of a facoal-ode wo-o bouday alue oblem Afouz ad e al ooed Homooy-eubao mehod fo a d of me-facoal Eoluo Equao I 3 Daz ad e al ued Sumudu afom mehod fo olg facoal dffeeal equao ad facoal Dffuo-Wae equao Alo eamay 3 oled Facoal Paal Dffeeal Equao by Ug Waele Oeaoal Mehod Meechae adea ad e al uggeed hee of fe dffeece aomao whch ae he mlc Eule mehod he elc Eule mehod ad he facoal Cac-cholo mehod

2 H Azz Gh Bad Loghma / J Mah Comue Sc fo FPDE baed o hfed Güwald fomula Alo hey deed ome dealed ably ad coegece aaly 8 9 he ace facoal dffuo equao ae a ye of facoal aal dffeeal equao whch by may auho ae oled umecally Fo eamle Khade 6 ued Chebyhe collocao mehod o ceze ace facoal dffuo equao o oba a lea yem of oday dffeeal equao ad ued he fe dffeece mehod fo olg he eulg yem Saadamad ad Dehgha 5 ued au aoach ad Soua 6 aled le o ole ace facoal dffuo equao I h ae we ooed a dffee aoach o oba he oluo of ace facoal dffuo equao u wh al codo ad bouday codo u < < < u g < < u h < 3 u h < 4 Whee he fuco a ouce em ad he facoal deae he Cauo deae ca be defed by 6: u m m oe ha fo Eq he clacal dffuo equao m u m < m m u u h wo ee a umecal mehod o ole h d of oblem ug fe dffeece cheme ad collocao mehod a Chebyhe olyomal Deco of he mehod I h eco he oce of olg he ace facoal dffuo equao decbed a -4 Le M whee M M F we ue a fe dffeece echque ad weghed cheme o cze he me deae u u d u d u hee Ug he oao u u 5 whee we oba u whee u d u d u M 6 5 7

3 H Azz Gh Bad Loghma / J Mah Comue Sc Defo 9 he well-ow Chebyhe olyomal of he f d of degee ae defed o he eal a co acco whee ad hey afy he ecuece elao: I ode o ue hee olyomal o he eal we defe he o called hfed Chebyhe olyomal by oducg he chage of aable z he hfed Chebyhe olyomal ae defed a: ow we ead u by hfed Chebyhe olyomal: u M 7 whee ae uow coeffce Fom equao 6 ad 7 we oba: M 8 I ode o fd he uow coeffce Chebyhe collocao mehod wh collocao o co ae aled M 9 Alo bouday codo 3 ad 4 fo M ae ued o oba: u h u h heefo eqauo 9- geeae a e of + algebac equao whch ca be oled fo uow coeffce Clealy u ca be obaed fom he al codo a follow: u u g 3 he ma fom of he ooed mehod I ode o fd he ma fom of uggeed mehod f by ug 9 fo we oba he elmay mace ad Q Fally by ug ad he ma fom fo h mehod Acheed Le u u u u ad whee u u 8

4 H Azz Gh Bad Loghma / J Mah Comue Sc Fom 7 we oba u whee a ma ha ge by Alo fom he gh de of equao 6 ad 9 fo we oba u d u d heefo he ma fom fo h equao a follow: Q Q whee ad he ma eleme of Q ae q fo Sce < * D ad fom 6 we hae:!!!! * D hu he ma eleme of Q ca be demoaed a follow: 3!!! 3! q oe ha * D fo hu q fo heefoe aboe mace ad equao 9 fo lea o he followg ma fom: Q Q ow by ug bouday codo ad elao we oba he ma fom of ou mehod a follow:

5 H Azz Gh Bad Loghma / J Mah Comue Sc whee he ma eleme of A ae: A B 3 a 3!! 3!! ma B obaed by addg wo zeo ow o he f ad la ow of ma Q ad 4 umecal eamle d d h h I h eco we code ome eamle of ooed cheme fo he ace facoal dffuo equao ad fo howg he accuacy effcecy ad aldy of he mehod we comae he ou umecal eul wh ohe meho Rema: I all of eamle he me e ae a ad Eamle 4 5 Code he followg ace facoal dffuo equao 5 u u 5 o a fe doma < < wh he dffuo coeffce 5 5 he ouce fuco wh he al codo u co ad he bouday codo u u fo > he eac oluo of h oblem u We aled he ee mehod wh ad comaed abolue eo fuco u u of ou cheme wh he mehod 5 whch ae how able I able ao he mamum abolue eo fo 9 ad < < ae eoed Alo fgue how he eac oluo ad aomae oluo fo u wh oe ha he mace A ad B fo h eamle ae a follow: A B 998 3

6 H Azz Gh Bad Loghma / J Mah Comue Sc Ealy ee ha A o-gula ma ad he ecal adu of ma A B heefoe he ooed mehod ha uque oluo ad ucodoally able 7 le ha oe able : Comao of ee mehod fo u wh he au mehod 5 fo Ea 4 Mehod5 wh m 7 ee mehod wh able : Abolue eo fo eamle 4 wh doma < < u u ao Fgue : Aomae oluo ad eac oluo fo u wh fo E 4 3

7 H Azz Gh Bad Loghma / J Mah Comue Sc Eamle 4 5 I h eamle we code he followg ace facoal dffuo equao: wh he al codo u 3 u ad he bouday codo u u fo > 8 8 u 3 8 e he eac oluo of h oblem u e We oled h equao by ug ooed mehod ad able 3 we comaed ou eul wh eul of obaed 6 5 Alo able 4 mamum abolue eo fo 4 8 ad < < ae eoed Alo fgue how he eac oluo ad aomae oluo fo u wh 3 oe ha he mace A ad B fo h eamle ae a follow: A B Ealy ee ha A o-gula ma ad he ecal adu of ma A B le ha oe heefoe he ooed mehod ha uque oluo ad ucodoally able able 3: Comao of abolue eo fo u fo eamle 4 Mehod 6 wh m5 Mehod 5 wh m5 ee mehod

8 H Azz Gh Bad Loghma / J Mah Comue Sc able 4: Abolue eo fo eamle 4 wh 3 doma < < u u ao Fgue : Aomae oluo ad eac oluo fo u wh 3 fo E 4 Eamle 43 6 Code he oblem wh al codo u 4 < < ad bouday codo u u e Le u e ad e he eac oluo fo h oblem We oled h oblem by alyg he ee mehod I able 5 we obaed he mamum eo bewee he eac oluo ad he aomae oluo fo dffee alue of fo 4 fe doma Alo fgue 3 how he eac oluo ad aomae oluo fo u wh 4 fo he mace A ad B fo h eamle fo ae a follow: 33

9 H Azz Gh Bad Loghma / J Mah Comue Sc A B Ealy ee ha A o-gula ma ad he ecal adu of ma A B le ha oe heefoe he ooed mehod ha uque oluo ad ucodoally able oe ha 6 h oblem ha bee oled by fe dffeece mehod ad le he 3 mamum eo fo 4 5 ad 8 wh ae ad 5 eecely able 5: he abolue eo fo 4 ad dffee alue of fo eamle Fgue 3: Aomae oluo ad eac oluo fo u wh 4 34

10 H Azz Gh Bad Loghma / J Mah Comue Sc Cocluo I h ae fe dffeece cheme ad Chebyhe collocao mehod hae bee uccefully aled o fd he oluo of he ace facoal dffuo equao Se ug ma fom of he mehod moe coee fo alcao of collocao mehod hu he ma fom of he ooed mehod wa obaed he eul ad comao of he ou ooed mehod ad ohe meho dcae ha h cheme accuae ad effce aoach fo he oluo of h oblem Pooed mehod fo he eamle of h ae wa ucodoally able bu ucodoally able fo he geeal cae oe oblem Refeece GA Afouz RA alaoh H Ahaga Elc aalycal oluo fo a d of me-facoal eoluo equao by He Homooy eubao meho J Mah Comu Sc JMCS Z Aazzadeh Z Beyg Rz FM Maale Gha GB Loghma A umecal oluo of olea aabolc-ye Volea aal egodffeeal equao ug adal ba fuco Eg Aal Boud Elem R Daz B Mohammadzade S Moua R Beheh Sumudu afom mehod fo olg facoal dffeeal equao ad facoal Dffuo-Wae equao J Mah Comu Sc JMCS K Dehelm he aaly of facoal dffeeal equao Bel Sge-Velag 5 GJ F JP Roo Lea quae fe eleme oluo of a facoal ode wo-o bouday alue oblem Comu Mah Al MM Khade O he umecal oluo fo he facoal dffuo equao Commu olea Sc ume Smul F Lu V Ah ad I ue umecal oluo of he ace facoal Foe-Pla equao J Comu Al Mah J Machado V Kyaoa ad F Maad Rce hoy of facoal caculu Commu olea Sc ume Smul JC Mao Dc Hacomb Chebyhe olyomal CRC e Boca Rao 3 MM Meechae C adea Fe dffeece aomao fo facoal adeco-eo flow equao J Comu Al Mah MM Meechae C adea Fe dffeece aomao fo wo-ded ace-facoal aal dffeeal equao Al ume Mah MM Meechae HP Scheffle C adea Fe dffeece meho fo wo-dmeoal facoal eo equao J Comu Phy A eamay B Aghel R Daz Solg facoal aal dffeeal equao by ug waele oeaoal mehod J Mah Comu Sc JMCS YRoh M Shoa Alcao of facoal calculu fo dyamc oblem of old mechac: oel e ad ece eul Al Mech Re A Saadamad M Dehgha A au aoach fo oluo of he ace facoal dffuo equao Comue Mah Al E Soua umecal aomao fo facoal dffuo equao a le Comue Mah Al H Swelam MM Khade AM agy umecal oluo of woded ace-facoal wae equao ug fe dffeece mehod J Comu Al Mah C adea MM Meechae H Scheffle A ecod-ode accuae umecal aomao fo he facoal dffuo equao J Comu Phy C adea MM Meechae A ecod-ode accuae umecal aomao fo he wo-dmeoal facoal dffuo equao J Comu Phy

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