Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Numercal soluon of second-order hyperbolc elegraph equaon va new cubc rgonomerc B-splnes approach Receved: May 17 Acceped: Augus 17 Frs Publshed: 3 Sepember 17 *Correspondng auhor: Muhammad Abbas, Deparmen of Mahemacs, Unversy of Sargodha, 1 Sargodha, Paksan E-mal: m.abbas@uos.edu.pk Revewng edor: Shaoyong La, Souhwesern Unversy of Fnance and Economcs, Chna Addonal nformaon s avalable a he end of he arcle Tahr Nazr 1, Muhammad Abbas 1 * and Muhammad Yaseen 1 Absrac: Ths paper presens a new approach and mehodology o solve he second-order one-dmensonal hyperbolc elegraph equaon wh Drchle and Neumann boundary condons usng he cubc rgonomerc B-splne collocaon mehod. The usual fne dfference scheme s used o dscreze he me dervave. The cubc rgonomerc B-splne bass funcons are ulzed as an nerpolang funcon n he space dmenson, wh a θ weghed scheme. The scheme s shown o be uncondonally sable for a range of θ values usng he von Neumann Fourer mehod. Several es problems are presened o confrm he accuracy of he new scheme and o show he performance of rgonomerc bass funcons. The proposed scheme s also compuaonally economcal and can be used o solve comple problems. The numercal resuls are found o be n good agreemen wh known eac soluons and also wh earler sudes. Subjecs: Compuer Mahemacs; Mahemacal Modelng; Mahemacal Physcs Tahr Nazr ABOUT THE AUTHORS Tahr Nazr s a PhD suden n Deparmen of Mahemacs, Unversy of Sargodha, Sargodha. He has obaned hs MPhl degree n Mahemacs from Unversy of Sargodha snce July 11 and maser s degree n Mahemacs from Deparmen of Mahemacs, Unversy of he Punjab, Lahore, Paksan. Hs research neress are Numercal mehods and splne appromaons. Muhammad Abbas s an asssan professor of Mahemacs a Unversy of Sargodha, Sargodha, Paksan. He compleed hs bachelor and masers from he Unversy of he Punjab, Lahore-Paksan n he years 1 and 3, respecvely. In 1, he obaned hs Docorae n Compuer Graphcs a School of Mahemacal Scences, Unvers Sans Malaysa, Penang, Malaysa. Hs research focus s n he area of Compuer Aded Graphc Desgn, Numercal mehods and splne appromaons. Muhammad Yaseen s an asssan professor of Mahemacs a Unversy of Sargodha, Paksan. He receved hs MSc and MPhl degrees from Quade-Azam Unversy Islamabad, Paksan. Hs area of neres s Numercal Analyss. He s currenly dong hs PhD from Unversy of Sargodha. PUBLIC INTEREST STATEMENT The rgonomerc B-splne funcons were used eensvely n Compuer Aded Geomerc Desgn CAGD as ools o generae curves and surfaces. An advanage of hese pecewse funcons s s local suppor properes where he funcons are sad o have suppor n specfc nerval. Due o hese properes, rgonomerc B-splnes have been used o generae he numercal soluons of lnear and non-lnear paral dfferenal equaons. In hs paper, he cubc rgonomerc B-splne bass funcon s consdered. Collocaon mehod based on he proposed bass funcons and fne dfference appromaon are developed o solve he one-dmensonal elegraph equaon. Trgonomerc B-splnes are used o nerpolae he soluon n -dmenson and fne dfference appromaons are used o dscreze he me dervaves. The proposed mehod has been proved o be uncondonally sable. 17 The Auhors. Ths open access arcle s dsrbued under a Creave Commons Arbuon CC-BY. lcense. Page 1 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Keywords: second-order one-dmensonal elegraph equaon; cubc rgonomerc B-splne bass funcons; cubc rgonomerc B-splne collocaon mehod; sably AMS subjec classfcaons: 65D7; 65N3; 65N35; 7S; A1; 1A5; A15; 65T Inroducon Problem Consder he second-order one-dmensonal hyperbolc elegraph equaon he elegraph equaon, gven by u u, +α, +β u, = u, +q, a b, wh nal condons 1 u, =g 1, u, =g, a b and he followng wo ypes of boundary condons 1 Drchle boundary condons ua, =f 1, ub, =f, 3 Neumann boundary condons u a, =w 1, u b, =w,. Applcaons The sudy of elecrc sgnal n a ransmsson lne, dspersve wave propagaon, pulsang blood flow n areres and random moon of bugs along a hedge are amongs a hos of physcal and bologcal phenomena whch can be descrbed by he elegraph Equaon Deals of he above-menoned phenomena and oher phenomena whch can be descrbed by he elegraph Equaon 1 can be found n Bohme 1987, Dehghan and Ghesma 1, Mohany & Jan, 1a and Pascal 1986. Clearly, he equaon and s soluon are of mporance n many areas of applcaons. 3. Leraure revew Several numercal mehods have been developed o solve he elegraph equaon subjec o Drchle boundary condons and he references are n Mohany and Jan, 1a, 1b, Mohany, Jan, and Arora, Mohany and Mohany, Jan, and George 1996. In Lu, Lu, and Chen 9, wo sem-dscrezaon mehods based on quarc splnes funcon have been developed o solve he elegraph equaons. A class of uncondonally sable fne dfference schemes consruced wh he help of quarc splnes funcons has been developed by H. W. Lu and L. B. Lu 9 for he soluon of he elegraph equaon. Furher several numercal mehods have been developed by Dehghan and Shokr 8 and Mohebb and Dehghan 8 n collaboraon wh dfferen auhors. These nclude he hn plae splnes radal bass funcons RBF for he numercal soluon of he elegraph equaon Dehghan & Shokr, 8 and hgh-order compac fne dfference mehod o solve he elegraph equaon Mohebb & Dehghan, 8. Furher deals on oher numercal mehods ncludng nerpolang scalng funcons Lakesan & Saray, 1, RBFs Esmaelbeg, Hossen, & Mohyud-Dn, 11, quarc B-splne collocaon mehod QuBSM Dos & Nazem, 1, cubc B-splne collocaon mehod CuBSM Mal & Bhaa, 13; Rashdna, Jamalzadeh, & Esfahan, 1 for he soluon of he elegraph equaon subjec o Drchle boundary condons Page of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 are n he leraure. Thus many numercal mehods have been developed o solve he elegraph Equaon 1 wh Drchle boundary condons. Some numercal mehods have been developed for numercal soluon of he elegraph equaon wh Neumann boundary condons. These nclude mehods by Dehghan and Ghesma 1 who consruced a dual recprocy boundary negral equaon DRBIE mehod n whch cubc radal bass funcon C-RBF, hn plae splne radal bass funcon TPS-RBF and lnear radal bass funcons L-RBF are ulzed for he numercal soluon of he elegraph equaon wh Neumann boundary condons. L. B. Lu and H. W. Lu 13 have developed a compac dfference uncondonally sable scheme CDS o solve he elegraph equaon wh Neumann boundary condons. Furher, Mal and Bhaa 1 have developed a echnque based on collocaon of cubc B-splne collocaon mehod CuBSM for solvng he elegraph equaon wh Neumann boundary condons. The rgonomerc B-splne collocaon mehod has araced aenon n he leraure and has been used for he numercal soluons of several lnear and non-lnear paral dfferenal equaons Abbas, Majd, Ismal, & Rashd, 1a, 1b, 1c; Zn, Abbas, Majd, & Ismal, 1; Zn, Majd, Ismal, & Abbas, 1a, 1b. The rgonomerc B-splnes have many geomerc properes lke local suppor, smoohness and capably of handlng local phenomena. There properes make rgonomerc B-splne approprae o solve lnear and non-lnear paral dfferenal equaons easly and efforlessly. Fyfe 1969 found ha he splne mehod s beer han he usual fne dfference scheme because has he flebly o oban he soluon a any pon n he doman wh greaer accuracy. The rgonomerc B-splne produced more accurae resuls for lnear and non-lnear nal boundary value problems as compared o radonal B-splne funcons Abd Hamd, Abd Majd, & Md Ismal, 1; Nkols, 1995. In hs work, a numercal collocaon fne dfference echnque based on cubc rgonomerc B-splne s presened for he soluon of elegraph Equaon 1 wh nal condons n Equaon and dfferen wo ypes of boundary condons n Equaons 3 and. Several sudes have been carred ou as he ordnary B-splne collocaon mehods o solve he proposed problem subjec o dfferen ypes of boundary condons bu no wh cubc rgonomerc B-splne collocaon mehod. A usual fne dfference scheme s appled o dscreze he me dervave whle cubc rgonomerc B-splne s ulzed as an nerpolang funcon n he space dmenson. The proposed mehod s uncondonally sable over.5 θ 1 and hs s proved by von Neumann approach. The feasbly of he mehod s shown by es problems and he appromaed soluons are found o be n good agreemen wh he eac soluons. The proposed mehod s superor o C-RBF Dehghan & Ghesma, 1, TPS-RBF Dehghan & Ghesma, 1, L-RBF Dehghan & Ghesma, 1, RBF Dehghan & Shokr, 8, QuBSM Dos & Nazem, 1, CDS L. B. Lu & H. W. Lu, 13, CuBSM Mal & Bhaa, 13, 1 due o smaller sorage and CPU me n seconds.. Oulnes of curren paper The oulne of hs paper s as follows: n Secon, he cubc rgonomerc B-splne collocaon mehod s eplaned. In Secon 3, numercal soluon of proposed problem 1 s dscussed. In Secon, he sably of proposed mehod s nvesgaed. In Secon 5, he resuls of numercal epermens are presened and compared wh eac soluons and some prevous mehods. Fnally, n Secon 6, he concluson of hs sudy s gven.. Descrpon of new rgonomerc B-splne mehod In hs approach, he space dervaves are appromaed usng cubc rgonomerc B-splne mehod CuTBSM. A mesh Ω whch s equally dvded by knos no N subnervals [, +1 ], =, 1,,, N 1 such ha, Ω:a = < 1 < < N = b s used. For he elegraph equaon 1, an appromae soluon usng collocaon mehod wh cubc rgonomerc B-splne s obaned n he form Abd Hamd e al., 1; Nkols, 1995 Page 3 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 N 1 U, = C TB = 3 where C are o be calculaed for he appromaed soluons u, o he eac soluons u ec,, a he pon, j. A C pecewse cubc rgonomerc B-splne bass funcons TB over he unform mesh can be defned as Abbas e al., 1a, 1b, 1c. TB = 1 ω where, ξ =sn ξ 3, [, +1 ] ξ ξ ζ + +ζ +3 ξ +1 + ζ + ξ +1, [ +1, + ] ζ + ξ +1 ζ +3 +ζ + ξ + + ξ ζ +3, [ +, +3 ] ζ 3 +, [ +3, + ], ζ =sn, ω = sn h sn h sn The appromaons U j a he pon, j over subnerval [, +1 ] can be defned as: 5 6 3h and h =b a N. U j = 1 k= 3 C j k TB k 7 The values of TB and s dervaves a knos are requred o oban he appromae soluons and hese dervaves are recorded n Table sn where a 1 = h snh sn 3h, a = 1+ cosh, a = 3 3 sn 3h, a = 3 sn 3h, a = 31+3 cosh 5 16 sn h cos h +cos 3h, 3 cos a 6 = h sn h 1+ cosh. From 5 and 6, he values a he knos of U j and her dervaves up o second order are calculaed n he erms of me parameers C j as: U j = a 1 C j + a 3 Cj + a 1 Cj 1 j U = a 3 C j + a 3 Cj 1 j U = a 5 C j + a 3 6 Cj + a 5 Cj 1 8 The Equaon 5 and boundary condons gven n 3 and are used o oban he appromae soluon a end pons of he mesh as: { U, j+1 =a 1 C 3 + a C + a 1 C 1 = f 1 j+1 U N, j+1 =a 1 C N 3 + a C N + a 1 C N 1 = f j+1 9 and { U, j+1 =a 3 C 3 + a C 1 = w 1 j+1 U N, j+1 =a 3 C N 3 + a C N 1 = w j+1 1 Table Values TB and s dervaves Bass +1 + +3 + TB a 1 a a 1 TB a 3 a TB a 5 a 6 a 5 Page of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 3. Numercal soluon of elegraph equaon In hs secon, a numercal soluon of elegraph Equaon 1 s obaned usng collocaon approach based on cubc rgonomerc bass funcons. The dscrezaon n me dervave s obaned by forward fne dfference scheme and θ weghed scheme appled o problem 1 o oban a r-dagonal of lnear equaons. The proposed θ weghed scheme s closely relaed o he accuracy of he mehod and numercal sably. A unform mesh Ω wh grd pons, j o dscreze he grd regon Δ =[a, b] [, T] wh = a + h, =, 1,,, N and j = jδ, j =, 1,, 3,, M, s used T = MΔ. The quanes h and Δ are mesh space sze and me sep sze, respecvely. Usng θ weghed echnque, he appromaons for he soluons of elegraph Equaon 1 a j+1 h me level can be gven by as Abbas e al., 1b U j + αu j = θ g j+1 +1 θg j + q, j θ [, 1] 11 where g j =U j β U j and he subscrps j and j + 1 are successve me levels, j =, 1,,, M. Usng he cenral fne dfference dscrezaon of he me dervaves and rearrangng he Equaon 11, we oban U j+1 U j + U j 1 Δ + α U j+1 U j Δ The Equaon 1 yelds as =1 θ g j + θ gj+1 + q, j j+1 1 + αk U k θ g j+1 = 1 + αku j + k 1 θ g j Uj 1 + k q, j 1 13 where k =Δ s he me sep. I s noed ha he sysem becomes an eplc scheme when θ =, a fully mplc scheme when θ = 1, and a Crank Ncolson scheme when θ = 1 Abbas e al., 1a, 1b. Hence, 13 becomes, 1 + αk U j+1 = 1 + αku j + k 1 θ k θ U j+1 U j β U j β U j+1 U j 1 + k q, j 1 The nal condon s subsued no las erm of Equaon 1 for compung C 1. By cenral dfference appromaon, U 1 = U 1 kg 15 Afer ha, he sysem hus obaned for j 1 on smplfyng 1 afer usng 8 consss of N + 1 lnear equaons n N + 3 unknowns C j+1 = C j+1, 3 Cj+1, Cj+1,, 1 Cj+1 a he me level = N 1 j+1. The boundary condons gven n Equaons 9 or 1 are used for wo addonal lnear equaons o oban a unque soluon of he resulng sysem. Thus, he sysem becomes a mar sysem of dmenson N + 3 N + 3 whch s a r-dagonal sysem ha can be solved by he Thomas Algorhm Burdern & Fares, ; Hoffman, 199; Iyengar & Jan, 9; Rosenberg, 1969; Sasry, 9. 3. Inal sae Afer he nal vecors C have been compued from he nal condons, he appromae soluons U j+1 a a parcular me level can be calculaed repeaedly by solvng he recurrence relaon 1 Abbas e al., 1a, 1b. C can be obaned from he nal and boundary values of he dervaves of he nal condon as follows Abbas e al., 1b. Page 5 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 U = g, 1 = U = g 1, =, 1,,, N U = g, 1 = N 16 Thus he Equaons 16 yeld a N + 3 N + 3 mar sysem for whch he soluon can be compued by he use of he Thomas algorhm.. Sably of proposed mehod In hs secon, he von Neumann sably mehod s appled o nvesgae he sably of he proposed scheme. Such an approach has been used by many researchers Abbas e al., 1a, 1b, 1c; Sddq & Arshed, 13. Subsung he appromae soluon U,, her dervaves a he knos wh q, = Srkwerda,, chaper 9, no Equaon 1 yelds a dfference equaon wh varables C m gven by 1 + αk + k θβ a 1 k θa 5 C j+1 + 1 + αk + k θβ a m 3 k θa 6 + 1 + αk + k θβ a 1 k θa 5 C j+1 m 1 = + αk 1 θk β a 1 +1 θk a 5 C j m 3 + + αk 1 θk β a +1 θk a 6 C j m + + αk 1 θk β a 1 +1 θk a 5 C j m 1 a 1 C j 1 + a m 3 Cj 1 + a m 1 Cj 1 m 1 C j+1 m 17 Smplfyng leads o w 1 C j+1 m 3 + w Cj+1 m + w 1 Cj+1 m 1 = w 3 Cj m 3 + w Cj m + w 3 Cj m 1 a 1 Cj 1 m 3 a Cj 1 m a 1 Cj 1 m 1 18 where w 1 =1 + αk + k θβ a 1 k θa 5, w =1 + αk + k θβ a k θa 6, w 3 = + αk 1 θk β a 1 +1 θk a 5, w = + αk 1 θk β a +1 θk a 6 19 Now on nserng he ral soluons one Fourer mode ou of he full soluon a a gven pon m, C j m = δj epm ηh no Equaon 18 and rearrangng he equaons, η s he mode number, h s he elemen sze and = 1, we oban w 1 δ j+1 e ηm 3h + w δ j+1 e ηm h + w 1 δ j+1 e ηm 1h = w 3 δ j e ηm 3h + w δ j e ηm h + w 3 δ j e ηm 1h a 1 δ j 1 e ηm 3h + a δ j 1 e ηm h + a 1 δ j 1 e ηm 1h Dvdng Equaon by δ j 1 e ηm h and rearrangng, we oban δ w + w 1 cos ηh δ w + w 3 cos ηh + a + a 1 cos ηh = 1 The wave number s gven as: η = π λ where λ s he wave lengh. Le Page 6 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 N = λ h whch represens he number of grd nerval over one wavelengh. Then he Equaon can be rearranged o he form Srkwerda, φ = η h = π N 3 where φ = η h s dmensonless wave number. As he shores waves represened a he consdered grd pons have wavelengh h, whereas he longes ones end o nfny, hen N mples ha φ π Srkwerda,. Le A = w + w 1 cos φ B = w + w 3 cos φ C = a + a 1 cos φ Then he Equaon 1 yelds A δ B δ + C = Applyng he Rouh Hurwz creron Sddq & Arshed, 13 on Equaon, he necessary and suffcen condons for Equaon 1 o be uncondonally sable as follows: Consder he ransformaon δ = 1+ξ and smplfyng he Equaon 1 becomes as 1 ξ A + B + Cξ + A Cξ + A B + C = 5 The uncondonally sably condon ξ 1 under he followng necessary and suffcen condons A + B + C, A C, A B + C 6 A + B + C = w + w + a + w1 + w 3 + a 1 cos φ A B + C = w w + a + w1 w 3 + a 1 cos φ A C = w a + w1 a 1 cos φ 7 Snce φ ranges from o π, hen nequales 6 can be verfy for s ereme values only Srkwerda,. Seng φ = π, he values of w, = 1,, 3, and a, = 1,, can be easly proved ha A + B + C = 161 + kα+k 3 + β 1 + θ sn h The nequaly gven n Equaon 8 sasfy f 1 + θ θ 1. cos ec h 8 A B + C = k cos ec h 6 cos h + 8β sn h A C = k cos ec h 6 k θ cos h + 8α + kβ θ sn h 9 3 Thus he proposed scheme for elegraph equaon s uncondonally sable n he regon.5 θ 1 whou any resrcon on grd sze and me sep sze bu h should be chosen n such a way ha he accuracy of he scheme s no degraded. Page 7 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 5. Numercal epermens Ths secon presens some numercal resuls of he hyperbolc elegraph equaon 1 wh nal and boundary condons 3 or. To es he accuracy of proposed mehod, several numercal epermens for dfferen values of α and β are gven n hs secon wh, L and roo mean square RMS errors are calculaed by = u ec U N = ma u J j U N j L = u ec U N = N h u j U N j j=o RMS = N u U j=o j N j. N + 1 We compare he numercal soluons obaned by cubc rgonomerc B-splne collocaon mehod for elegraph equaon 1 wh known eac soluons and hose numercal mehods n he leraure. We carry ou 1 by he proposed mehod and Inel Core TM 5-1M CPU@.3 GHz wh 8GB RAM and 6-b operang sysem Wndows 7. The numercal mplemenaon s carred ou n Mahemaca 9. Numercal resuls are compued by cubc rgonomerc B-splne collocaon mehod for he elegraph equaon 1 a dfferen me levels wh smaller sorage and CPU me whch are abulaed n dfferen Tables. All Fgures are drawn a he value of weghng parameer θ =.5. Eample 1 Consder he followng parcular case of Equaon 1 n he doman [, π ] wh α =, β = Dehghan & Shokr, 8; Mal & Bhaa, 13 u u, +, +u, = u, +q, π, subjec o he followng nal and boundary condons { u, = =sn, u, = = sn u =, =, u = π, = where q, = e sn. The eac soluon of hs problem s u ec, =e sn. The proposed mehod s appled o calculae he numercal soluons of he elegraph equaon 1 wh h =., Δ =.1 a dfferen me levels. The absolue errors and relave error L a weghng parameer θ =.5, dfferen me levels and also CPU me n second, are repored n Table. I can be concluded ha our resuls are more accurae as compared o resuls obaned by Dehghan and Shokr 8 and Mal and Bhaa 13. In Table 3 and Fgure 1, we repor he absolue errors, relave errors and RMS for h =., Δ =.1 a dfferen me levels Table. Relave errors, mamum errors and CPU me of Eample 1 wh Δ =.1, h =. CuTBSM CuBSM Mal & Bhaa, 13 RBF Dehghan & Shokr, 8 L θ CPU s L CPU s L CPU s.5.5e-8 3.55E-8.56.89.33E-6 86E-6 3. 7.95E-5 8.37E- 6 5 5.E-9.1E-9.56.15.36E-6 3.8E-6.89 5E- 57E- 5 5 8.36E-1 6.66E-1.56 5.1.78E-6 3.8E-6 5.7 59E- 7E- 5. 8.6E-1 6.E-1.56 7..7E-6 3.E-6 7.53 E- 58E- 5 1 19 8 Page 8 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Table 3. Relave errors, mamum errors, RMS, values of θ and CPU me of Eample 1 wh Δ =.1, h =. CuTBSM CuBSM Mal & Bhaa, 13 L RMS θ CPU s L CPU s.5 7E-7 8.56E-8 6.3E-8.665.3.33E-6 86E-6.51.56E-8.E-8 E-8.66615.5.36E-6 3.8E-6.59 5.38E-8 9E-8 33E-8.6669.56.78E-6 3.8E-6.63. 17E-8 9.38E-9 6.61E-9.6666.65.7E-6 3.E-6.68 Fgure Error graph of Eample 1 a dfferen me levels wh h =., Δ =. 8. 1 8 Errors 6. 1 8. 1 8.5 5.. 1 8.5 5..5 3. wh CPU me wh dfferen values of weghng parameer θ due o he purpose of comparson wh esng mehods. The numercal resuls of hs problem are n good agreemen wh eac soluon and are more accurae han cubc B-splne collocaon mehod Mal & Bhaa, 13. Fgure depcs he graphs of comparson beween eac and numercal soluons a me levels = 1,, 3 wh h =., Δ =. Fgure 3 shows he space me graph of eac and appromae soluons a = 3 wh h =., Δ =. Eample In hs problem, we consder he elegraph equaon 1 n he doman [, ] wh α = 1, β = 5 Dos & Nazem, 1; Mal & Bhaa, 13 u u, +, +5u, = u, +q,, subjec o he followng nal and boundary condons u, =an u, 1,= 1 + an u, =an, u, =an + and funcon q, =α 1 + an + + β an +. The eac soluon of hs equaon s u ec, =an +. In hs problem, we ake L =, h =. and wo values of me sep sze k =.1 and k =.1 due o he purpose of comparson wh esng mehods. In Table, we repor he absolue errors and relave errors of hs problem usng presen mehod a dfferen me levels and dfferen values of weghng parameer θ. In Table 5, we also recorded he absolue errors and relave errors a dfferen me levels for h =.1, k =.1 and concluded ha our resuls are more Page 9 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Fgure. Comparson of numercal and eac soluon of Eample 1 a dfferen me levels wh h =., Δ =. U,.35.3.5..15.1 5..5 3..5 5..5 3. Fgure 3. Space me graphs of Eample 1 a T = 3 wh h =., Δ =..8 Space Tme graph of Appromae soluon a 3. Space Tme graph of Eac soluon a 3..8.6 U,.. U,.6...8 6.. 3. 3..5 5..5 3...8 6.. 3. 3..5 5..5 3 Table. L, errors of Eample a dfferen me levels wh h =. Mehod L k =.1 k =.1 L k =.1 k =.1 CuTBSM..13E-5 3.35E-5 85E-.79E-. 5.5E-5 7.9E-5.3E- 6.35E-.6 E- 63E- 7.96E- 17E-3.8.39E-.65E- 7E-3.E-3 73E-3 5.59E-3.19E-3 9.51E-3 CuBSM Mal & Bhaa, 13. 5.3E-5 3.8E-5 88E-.63E-. 9.5E-5 5.3E-5.89E- 7.E-.6.E- 9.7E- 9.9E- 9E-3.8 7.83E- 88E- 87E-3 3.1E-3 7.9E-3 5.87E- 5.1E-3 1E- Table 5. Relave errors and mamum errors of Eample wh Δ = h =.1 CuTBSM CuBSM Mal & Bhaa, 13 QuBSM Dos & Nazem, 1 L L. 8E-.7E-.18E- 3.61E-.77E-..3E- 6.17E- 5.66E- 3E- 7.8E-.6 7.71E- 1E-3 15E-.59E-3 39E-3.8 37E-3.18E-3,6E-3 7.6E-3 3.9E-3.99E-3 5.63E-3 3E-.66E- 3E-3 Page 1 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Fgure. Numercal and eac soluons of Eample a dfferen me levels wh h =., Δ =. U, 1 1 1 8.8.6.. 6.5 5. Fgure 5. 3D soluons plo of Eample a = wh h =.. Space Tme graph of Appromae soluona Space Tme graph of Eac soluona 1. 1. 8. 8. U, 6. U, 6......9.8.7.6.5..3..1...6.8. 8 6.9.8.7.6.5..3..1...6.8. 8 6 accurae han Dos and Nazem 1 and Mal and Bhaa 13. Fgure llusraes he comparson of eac soluon wh appromae soluon of hs problem a varous me levels and dfferen values. In Fgure 5, we show he space me graph of appromae and eac soluons a me =. Eample 3 We consder he elegraph equaon 1 n he doman [, 1 ] wh α =.5, β = Dehghan & Shokr, 8; Mal & Bhaa, 13 u u, +, +u, = u, +q, subjec o he followng nal and boundary condons { u,=, and u, = 1&u, =, u1, = q, = + e + e. The eac soluon of hs problem s u ec, = e. The absolue errors, relave errors and CPU me n seconds s shown n Table 6 wh Δ =.1, h =. Numercal resuls are compared wh he obaned resuls n Dehghan and Shokr 8 and Mal and Bhaa 13. I can be concluded ha he numercal soluons obaned by our mehod are good n comparson wh Dehghan and Shokr 8 and Mal and Bhaa 13. The graph of eac and numercal soluons a = 1,, 3,, 5 s shown n Fgure 6 and he space me graph of soluons up o = 5 s presened n Fgure 7. Page 11 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Table 6. Relave errors, mamum errors and CPU me of Eample 3 wh Δ =.1, h =.1 CuTBSM CuBSM Mal & Bhaa, 13 RBF Dehghan & Shokr, 8 L CPU s L CPU s L CPU s 6.31E-5 8.76E-5.3.55E-5 5.91E-5.3 E- 85E-6..3E-5 3.9E-5.57 3E-5 78E-5.77 8.8E-5 7E-5 3..6E-6 5.9E-6 5 6.E-6 3E-5 15 9E- 8E-5 1..19E-5 3.E-5 11 8.9E-6 35E-5 9 18E- 65E-5 1 5. 5.18E-6 6.9E-6 6 3.1E-6 5.E-6 6 7.55E-5 5E-5 Fgure 6. Comparson of numercal and eac soluons of Eample 3 a dfferen me levels wh h =.1, Δ =. U,.1.1.1.8.6. 3.... 5....6.8 Fgure 7. Space me surface plo of soluons for Eample 3 a = 5. wh h =..1.1 Eac Tme graph of Appromae soluona 5..1.1 Eac TmegraphofEac soluon a 5..1.1 U,.8 U,.8.6.6.....5 5..5 3. 3.5..5 5..9.8.7.1..3..5.6.5 5..5 3. 3.5..5 5..9.8.7.1..3..5.6 Eample Consder he elegraph equaon 1 n he doman [, 1 ] and α = 6, β = Dos & Nazem, 1; Mal & Bhaa, 13 u u, +1, +u, = u, +q, wh followng nal and boundary condons { u, =sn, u, = 1 u, =, u1, =cos sn1 and q, = α sn sn+β cos sn. The eac soluon of hs problem s u, =cos sn. The effcency can be noed from Table 7 usng L, and RMS errors wh Δ =.1, h =. In Table 8, we also repored he absolue errors and relave errors a dfferen me levels for Page 1 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Table 7. Relave errors, mamum errors and RMS errors of Eample wh Δ =.1, h =.1 CuTBSM CuBSM Mal & Bhaa, 13 L RMS L RMS..96E-6.63E-6.9E-6.69E-6 5.E-6.67E-6. 6.77E-6 1E-5 6.73E-6 5.61E-6 8.61E-6 5.59E-6.6 9.81E-6 E-5 9.76E-6 9.75E-6 5E-5 9.7E-6.8 E-5 71E-5 19E-5 38E-5.3E-5 37E-5 3E-5 9E-5 33E-5 73E-5.75E-5 7E-6 Table 8. L and errors of Eample a dfferen me levels wh k =.1 Mehod L h =.1 h =.1 L h =.5 h =.5 CuTBSM..9E-5.56E-5.9E-5.56E-5. 6.69E-5 9.99E-5 6.68E-5 9.98E-5.6 9.7E-5 E- 9.69E-5 E-.8 19E- 7E- 18E- 7E- 33E- 88E- 88E- 88E- CuBSM Mal & Bhaa, 13. 3.67E-5 7.91E-5 3.3E-5 6.8E-5. 8.9E-5 6E- 8.57E-5 9E-.6 37E-.3E- 33E-.E-.8 79E-.98E- 75E-.89E-.13E- 3.51E-.9E- 3.3E- QuBSM Dos & Nazem, 1..3E-5. 7.93E-5.6 1E-.8 9E- 65E- dfferen values h wh Δ =.1 and he numercal resuls are compared wh hose of Dos and Nazem 1 and Mal and Bhaa 13. We found ha our numercal resuls are comparable o ha of QuBSM Dos & Nazem, 1 and CuBSM Mal & Bhaa, 13 n erms of L, errors. Fgure 8 presens he comparson of numercal and eac soluons for dfferen me levels wh Δ =.1, h =. The space me graph of numercal and eac soluons a = s presened n Fgure 9. Eample 5 Consder he followng parcular case of second-order one-dmensonal equaon 1 over he regon [, π ] [, 3 ] wh α =, β = Dehghan & Ghesma, 1; L. B. Lu & H. W. Lu, 13; Mal & Bhaa, 1 Fgure 8. Comparson of numercal and eac soluons of Eample a dfferen me levels wh h =.1, Δ =. U,.8.6....6.8....6.8 Page 13 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Fgure 9. 3D plo of appromae and eac soluons for Eample a = 5. wh h =..75.5.5 U,.5 Space Tme graph of Appromae soluona 5. U,.75.5.5.5 Space Tme graph of Eac soluona 5..5.5.75.75.5 5..5 3. 3.5..5 5..9.8.7.1..3..5.6.5 5..5 3. 3.5..5 5..9.8.7.1..3..5.6 u u, +8, +u, = u, +q, π, subjec o he followng nal and Neumann boundary condons { u, = =sn, u, = = sn u, =e, u π, =e where q, = e sn. The eac soluon of hs problem s u ec, =e sn. The proposed mehod s appled o calculae he numercal soluons of elegraph equaon 1 and a = 1,, 3 wh Δ =.1 and dfferen values of h. The absolue errors, relave errors and RMS errors a dfferen values of weghng parameer θ and also CPU me n second, are repored n Table 9. I can be concluded ha our resuls are more accurae as compared o resuls Table 9. L, and RMS errors and CPUs of Eample 5 a dfferen me levels wh k =.1 Mehod h θ L RMS CPU s CuTBSM.5.797 36E-6 7.E-7 5.3E-7...867 36E-6 7.39E-7 5.1E-7.51.5..797 3.E-7 93E-7 E-7.5..867.96E-7 89E-7 18E-7 3.5 3..797 8.33E-8 7.16E-8 3.31E-8.67..867 8.1E-8 6.87E-8 3.E-8 55.1.8117 8.13E-8 6.57E-8 3.1E-8.9 C-RBF Dehghan & Ghesma, 1.5 3. 7.1E-5. 71E-5.1 8.E-5 TPS-RBF Dehghan & Ghesma,.5 3. 9.1E-5 1..9E-5.1 8.99E-6 L-RBF Dehghan & Ghesma, 1.5 3. 3.1E-. 7.13E-5.1.3E- CuBSM Mal & Bhaa, 1.5 5.11E-.95E-.3E-. 78E- 67E- 7.11E-5.5. 3.E-.5E- 7E-. 53E- 7E- 6.11E-.5 3. 99E- 3E- 7.85E-.7. 7E- 6.61E-5.6E-5 7.1 3.76E-5 3.1 CDS L. B. Lu & H. W. Lu, 13.5 3. 6.E-7. 6.1E-7.1 6.E-7 Page 1 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Fgure Error graph of Eample 5 a dfferen values of h a = 3. wh Δ =. Errors 6. 1 8. 1 8. 1 8 h.5 h. h.1 1 3 5 6. 1 8. 1 8 6. 1 8 Fgure 1 Space me graph of soluons for Eample 5 up o = 3. wh k =.1 and h =.5. U,.5. Space Tme graphofappromae soluona 3. U,.5. Space Tme graphof Eac soluon a 3..5.5 3 6 3 6 1 1 obaned by hree RBFs schemes such as Cubc RBF CRBF Dehghan & Ghesma, 1, Thn Plae Splne RBF TPS-RBF Dehghan & Ghesma, 1, Lnear RBF L-RBF Dehghan & Ghesma, 1, CDS L. B. Lu & H. W. Lu, 13 and CuBSM Mal & Bhaa, 1. Fgure 1 depcs he errors of proposed mehod a dfferen values of h. The numercal resuls of hs problem are also n good agreemen wh eac soluon. Fgure 11 shows he space me graph of appromae and eac soluons a = 3 wh h =.5, Δ =. 6. Concluson Ths paper has nvesgaed he applcaon of cubc rgonomerc B-splne collocaon mehod o fnd he numercal soluon of he elegraph equaon wh nal condon and Drchle as well as Neumann s ype boundary condons. A usual fne dfference approach s used o dscreze he me dervaves. The cubc rgonomerc B-splne s used for nerpolang he soluons a each me. The numercal resuls shown n Tables 9 and Fgures 1 11 ndcae he relably of resuls obaned. The obaned soluon o he elegraph equaon for varous me levels has been compared wh he eac soluon and esng mehods by calculang, L and RMS errors. The comparson ndcaed mproved accuracy compared o C-RBF Dehghan & Ghesma, 1, TPS-RBF Dehghan & Ghesma, 1, L-RBF Dehghan & Ghesma, 1, RBF Dehghan & Shokr, 8, QuBSM Dos & Nazem, 1, CDS L. B. Lu & H. W. Lu, 13, CuBSM Mal & Bhaa, 13, 1. Page 15 of 17
Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Acknowledgemens The auhors are ndebed o he anonymous revewers for her helpful, valuable commens and suggesons n he mprovemen of hs manuscrp. Fundng The auhors receved no drec fundng for hs research. Auhor deals Tahr Nazr 1 E-mal: ahrnazr666@gmal.com ORCID ID: hp://orcd.org/-1-99-966 Muhammad Abbas 1 E-mal: m.abbas@uos.edu.pk ORCID ID: hp://orcd.org/--91-158 Muhammad Yaseen 1 E-mal: yaseen@uos.edu.pk ORCID ID: hp://orcd.org/-1-7999-87 1 Deparmen of Mahemacs, Unversy of Sargodha, 1 Sargodha, Paksan. Caon nformaon Ce hs arcle as: Numercal soluon of second-order hyperbolc elegraph equaon va new cubc rgonomerc B-splnes approach, Tahr Nazr, Muhammad Abbas & Muhammad Yaseen, Cogen Mahemacs 17, : 1386 References Abbas, M., Majd, A. A., Ismal, A. I. M., & Rashd, A. 1a. Numercal mehod usng cubc B-splne for a srongly coupled reacon-dffuson sysem. PLOS ONE, 91, 1 e8365. Abbas, M., Majd, A. A., Ismal, A. I. M., Rashd, A. 1b. 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Nazr e al., Cogen Mahemacs 17, : 13861 hps://do.org/18/3311835.17.13861 Journal of he Egypan Mahemacal Socey,, 9 13. Srkwerda, J. C.. Fne dfference schemes and paral dfferenal equaons nd ed.. Phladelpha, PA: Socey for Indusral and Appled Mahemacs. Zn, S. M., Abbas, M., Majd, A. A., & Ismal, A. I. M. 1. A new rgonomerc splne approach o numercal soluon of generalzed nonlnear Klen-Gordon equaon. PLOS ONE, 95, 1 e9577. Zn, S. M., Majd, A. A., Ismal, A. I. M., & Abbas, M. 1a. Cubc rgonomerc B-splne approach o numercal soluon of wave equaon. Inernaonal Journal of Mahemacal, Compuaonal, Physcal and Quanum Engneerng, 8, 11 116. Zn, S. M., Majd, A. A., Ismal, A. I. M., & Abbas, M. 1b. Applcaon of hybrd cubc B-splne collocaon approach for solvng a generalzed nonlnear Klen-Gordon equaon. Mahemacal Problems n Engneerng, 1, 1 pages. Arcle ID 1856. 17 The Auhors. Ths open access arcle s dsrbued under a Creave Commons Arbuon CC-BY. lcense. You are free o: Share copy and redsrbue he maeral n any medum or forma Adap rem, ransform, and buld upon he maeral for any purpose, even commercally. The lcensor canno revoke hese freedoms as long as you follow he lcense erms. Under he followng erms: Arbuon You mus gve approprae cred, provde a lnk o he lcense, and ndcae f changes were made. You may do so n any reasonable manner, bu no n any way ha suggess he lcensor endorses you or your use. No addonal resrcons You may no apply legal erms or echnologcal measures ha legally resrc ohers from dong anyhng he lcense perms. Cogen Mahemacs ISSN: 331-1835 s publshed by Cogen OA, par of Taylor & Francs Group. Publshng wh Cogen OA ensures: Immedae, unversal access o your arcle on publcaon Hgh vsbly and dscoverably va he Cogen OA webse as well as Taylor & Francs Onlne Download and caon sascs for your arcle Rapd onlne publcaon Inpu from, and dalog wh, eper edors and edoral boards Reenon of full copyrgh of your arcle Guaraneed legacy preservaon of your arcle Dscouns and wavers for auhors n developng regons Subm your manuscrp o a Cogen OA journal a www.cogenoa.com Page 17 of 17