A Computational Quadruple Laplace Transform for the Solution of Partial Differential Equations

Transcription

1 Appled Mathematcs,, 5, Publshed Onlne December n ScRes A Computatonal Quadruple Laplace Transform for the Soluton of Partal Dfferental Equatons Hamood Ur Rehman, Muammal Iftkhar, Shoab Saleem, Muhammad Youns, Abdul Mueed 3 Department of Mathematcs, Unversty of Educaton, Okara Campus, Okara, Pakstan Center for Undergraduate Studes, Unversty of the Punjab, Lahore, Pakstan 3 Ar Unversty Multan Campus, Multan, Pakstan Emal: hamood8@gmalcom, muamlftkhar@ueedupk, shaby55@yahoocom, younspu@gmalcom, abdulmueed3@hotmalcom Receved October ; revsed November ; accepted 6 November Copyrght by authors and Scentfc Research Publshng Inc Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY Abstract In ths paper, we proposed new results n quadruple Laplace transform and proved some propertes concerned wth quadruple Laplace transform We also developed some applcatons based on these results and solved homogeneous as well as non-homogeneous partal dfferental equatons nvolvng four varables The performance of quadruple Laplace transform s shown to be very encouragng by concrete examples An elementary table of quadruple Laplace transform s also provded Keywords Quadruple Laplace Transform, Exact Soluton, Convoluton, Partal Dfferental Equaton, Homogeneous and Non-Homogeneous Problems Introducton Many engneerng and scence felds encounter lnear or non-lnear partal dfferental equatons descrbng the physcal phenomena A number of methods (for example, approxmate and exact methods can be used to determne the solutons of dfferental equatons Mostly, t may be complcated to solve these equatons analytcally Such equatons are commonly solved by ntegral transforms such as Laplace and Fourer transforms and the worth of Laplace and Fourer transforms les n ther ablty to transform dfferental equatons nto algebrac How to cte ths paper: Rehman, HU, Iftkhar, M, Saleem, S, Youns, M and Mueed, A ( A Computatonal Quadruple Laplace Transform for the Soluton of Partal Dfferental Equatons Appled Mathematcs, 5,

2 H U Rehman et al equatons, whch allows a systematc and smple way to fnd soluton The numercal methods can provde approxmate solutons rather than analytc solutons of the problems [] A number of aspects of these methods have been studed n [] [3] The Laplace transform has been effectvely used to solve lnear and non-lnear ordnary and partal dfferental equatons and s used extensvely n electrcal engneerng The Laplace transform reduces a lnear dfferental equaton to an algebrac equaton, whch can be solved by rules of algebra The orgnal dfferental equaton can then be solved by applyng the nverse Laplace transform The Heavsde frst proposed a scheme, wthout usng the Laplace transform (see [] and references theren Eltayeb and Klçman [5] appled double Laplace transform to fnd the soluton of general lnear telegraph and partal ntegro-dfferental equatons Dahya and Najafa [6] establshed new theorems for calculatng the Laplace transforms of n-dmensons and applcaton of these theorems to a number of commonly used specal functons was consdered, and n the end, authors solved one-dmensonal wave equaton nvolvng specal functons usng two dmensonal Laplace transforms Aghl and Moghaddam [7] presented a new theorem and corollary on mult-dmensonal Laplace transformatons Authors further developed some applcatons based on these results Klçman and Eltayeb [8] dscussed the relatonshp between Sumudu and Laplace transforms and further made some comparson on the solutons Authors provded some counter examples Chenguel and Reghoua [9] nvestgated the soluton of three-dmensonal dffuson equaton wth non-local condton usng Adoman decomposton method Atangana [] ntroduced the trple Laplace transform Author dscussed some propertes and theorems about the trple Laplace transform Moreover, author used the operator to solve some knd of thrd-order dfferental equaton The am of ths paper s to dscuss some propertes and theorems about the quadruple Laplace transform and gve a good strategy for solvng the fourth order partal dfferental equatons n engneerng and physcs felds, by quadruple Laplace transform Frst of all, we recall the followng defntons f xy, can be defned [5] [] as The double Laplace transform of a contnuous functon xy px qy where x, y > and p, q are Laplace varables The nverse double Laplace transform s defned as F pq, L f xy, e e f xy, dd, xy ( xy f ( xy, L F xy ( pq α+ px β+ qy xy xy, e e (, d d π π F pq q p α β ( The trple Laplace transform [] of a contnuous functon f( xy,, can be defned as xy px qy r F ( pqr,, L f( xy,, e e e f( xy,, ddd, xy (3 where x, y, > and p, q, r are Laplace varables The nverse trple Laplace transform s defned as xy f xy L F pqr F pqr r q p π α π β π ξ xy α+ px β+ qy ξ+ r xy (,, xy (,, e e e (,, d d d ( In the followng secton defntons of quadruple Laplace transform, ts nverse and some of ts propertes are presented Defntons and Propertes Before launchng nto the man part of the paper, we defne some notatons and termnologes whch wll reman standard F ( pqrs,,, L f(,,,, wxy F ( pqr,,, Lwxy f(,,,, wx F ( pqy,,, Lwx f(,,,, w F pxy,,, L f,,, w 3373

3 H U Rehman et al Quadruple Laplace transform: Let f be a contnuous functon of four varables, then, the quadruple Laplace f,,, s defned by transform of pw qx ry s F pqrs,,, L f,,, e e e e f,,, d ddd, ( where w, x, y, > and p, q, r, s are Laplace varables It s to be noted that the quadruple Laplace transform operator s lnear { } { } { } L af w, x, y, + bg w, x, y, al f w, x, y, + bl g w, x, y, ( Now, f the quadruple Laplace transform s known, ts nverse s gven by (,,, (,,, f wxyt L F pqrs α+ pw β+ qx γ+ ry δ+ s e e e e F ( pqrs,,, d dy dx d w π α π β π γ π δ Quadruple Laplace transform for some partal dervatves of functon of four varables are gven as Quadruple Laplace transform for frst order partal dervatve of functon of four varables wxy L f(,,, sf ( pqrs,,, F ( pqr,,, Quadruple Laplace transform for second order partal dervatve of functon of four varables xy xy L f (,,, pf ( pqrs,,, pf (, qrs,, F (, qrs,,, w w (3 ( (5 wy wy L f (,,, qf ( pqrs,,, qf ( p,, rs, F ( p,, rs,, x x (6 wx wx L f (,,, rf ( pqrs,,, rf ( pq,,, s F ( pq,,, s y (7 y 3 Quadruple Laplace transform for the mxed fourth order partal dervatve of functon of four varables L f(,,, wxy wx wy pqrsf ( p, q, r, s pqrf ( p, q, r, pqsf ( p, q,, s prsf ( p,, r, s xy wx wy w qrsf (, q, r, s + pqf ( p, q,, + prf ( p,, r, + psf ( p,,, s xy x y w + qrf (, q, r, + qsf (, q,, s + rsf (,, r, s pf ( p,,, x qf (,,, y q rf (,, r, sf (,,, s + f (,,, (8 Lwwxx f (,,, w x wwxx x w (,,, (,,, (,,, w x (,,, w (,,, x (,,, (,,, (,,, (,,, pqf pqy pqf qy pqf p y + pqf y pf p y qf qy + pf p y + qf q y + f x w wx The unqueness and exstence of the quadruple Laplace transform s dscussed n the followng secton (9 3 Unqueness and Exstence of the Quadruple Laplace Transform Consder f(,,, be a contnuous functon on the nterval [, Also, assume that f(,,, s of exponental order, that s, there exsts some constants a, b, c, d R such that f(,,, satsfy the fol- 337

4 H U Rehman et al lowng condton The quadruple Laplace transform,,, > (,,, f sup < (3 aw+ bx+ cy+ d e pw qx ry s F p, q,, r s L f w, x, y, e e e e dwddd, x y (3 exsts for all p > a, q > b, r > c, s > d and satsfy the condton (3 The followng theorem explans the unqueness Theorem Let f(,,, and g(,,, s defned be contnuous functons defned for w, x, y, and havng Laplace transforms, F ( pqrs,,, and G ( pqrs,,, respectvely If F ( pqrs,,, G ( pqrs,,,, then f(,,, g(,,, Proof If α, β, γ, δ are suffcently large Then, from the defnton of the quadruple nverse Laplace transform, we have f F pqrs y x w π α π β π γ π δ α+ pw β+ qx γ+ ry δ+ s (,,, e e e e (,,, d d d d (33 Usng the hypothess, F ( pqrs,,, G ( pqrs,,,, the expresson (33 can be wrtten as f G pqrs y x w (,,, α+ pw β+ qx γ+ ry δ+ s e e e e (,,, d d d d π α π β π γ π δ (3 g(,,,, ths completes the proof Convoluton Theorem for the Quadruple Laplace Transform In ths secton, we wll gve some defntons of convoluton for functons and state the convoluton theorem of the quadruple Laplace transform Consder the functons f (,,,, f (,,,, f3 (,,, and f (,,, The convoluton of functons f and f can be defned as w x y f f f w ρ, x ϕ, y ψ, η f ρϕψ,,, η dρdϕdψd η ( The convoluton of functons f, f and f 3 can be defned as w x y ( ( ρ, ( ϕ, ( ψ, ( + f ( w x y f f f f3 f w w+ x x+ y y+ ( η ρ, ϕ, ψ, η ρϕψ,,, η dρdϕdψd η 3 Smlarly, the convoluton of functons f, f, f 3 and f can be defned as w x y ( ( ( ρ, ( ϕ, ( y+ y + ψ ( + + η f( w ( w + ρ x ( x + ϕ y ( y + ψ, ( + η f3( w ρ, x ϕ, y ψ, η f ( ρϕψ,,, η dρdϕdψd η f f f f f w w + w + x x + x + y 3 Theorem (Convoluton Theorem If,,, F p, q,, r s e e e e f w, x, y, dwddd, x y (3 pw qx ry s F p, q,, r s e e e e f w, x, y, dwddd, x y ( pw qx ry s 3375

5 H U Rehman et al F p, q,, r s e e e e f w, x, y, dwddd, x y (5 pw qx ry s 3 3 are convergent at the pont ( pqrs,,, and f F p, q,, r s e e e e f w, x, y, dwddd, x y (6 pw qx ry s s absolutely convergent, then, the followng expresson F pqrs,,, F pqrs,,, F pqrs,,, F pqrs,,, F pqrs,,,, (7 s the Laplace transform of the functon and the ntegral 3 (,,, w x y ( ( + + ρ, ( + + ϕ, ( y+ y + ψ ( + + η f( w ( w + ρ x ( x + ϕ y ( y + ψ, ( + η f3( w ρ, x ϕ, y ψ, η f ( ρϕψη,,, dρdϕdψd η, f f w w w x x x y,,, (8 pw qx ry s F p, q,, r s e e e e f w, x, y, dwddd, x y (9 s convergent at the pont ( pqrs,,,, see the proof n [] 5 Propertes of Quadruple Laplace Transform In ths sectoon, some propertes of quadruple Laplace transform are presented Property ( ( aw bx cy d F p aq, br, cs, d L e e e e f wx,, y, pqrs,,, Proof: By defnton ( of quadruple Laplace transform left hand sde of (5 can be solved as aw bx cy d L e e e e f,,, pqrs,,, ( aw bx cy d pw qx ry s e e e e e e e e f,,, d ddd aw bx cy pw qx ry d s e e e e e e e e f w, x, y, d dwdd x y The nner ntegral wth respect to n Equaton (5 can be solved after proper substtuton as Equaton (5 becomes (5 (5 e d e s s d f(,,, d e + f(,,, d F ( wxys,,, + d (53 aw bx cy d L e e e e f,,, pqrs,,, ( aw bx cy pw qx ry e e e e e e F wxys,,, + d dwxy dd aw bx pw qx cy ry e e e e e e F wxys,,, + d dy dwx d The nner ntegral wth respect to y n Equaton (5 can be solved after proper substtuton as e cy e ry (,,, d e c r y y F wxys + d y + F ( wxys,,, + d d y F ( wxc,, + rs, + d (55 After substtutng the value from Equaton (55 n Equaton (5, we can get aw bx cy d aw bx pw qx y L e e e e f,,, pqrs,,, e e e e F wxr,, + cs, + d dwx d (56 On the smlar lnes, ntegratng Equaton (56 twce accordngly, we can obtan requred result Equaton (5 3376

6 H U Rehman et al (5 Property ( p q r s F,,, L f ( αw, βx, γ y, δ ( p, q, r, s αβγδ α β γ δ Proof: By defnton ( of quadruple Laplace transform rght hand sde of (57 can be solved as L f w x y pqrs ( α, β, γ, δ (,,, ( α β γ δ ( ( α β γ δ pw qx ry s e e e e f w, x, y, d ddd pw qx ry s e e e e f w, x, y, d dwdxd y The nner ntegral wth respect to can be solved, after proper substtuton as (57 (58 e s f ( αw, βx, γ y, δ d F w, x, y, s δ δ (59 On the smlar lnes, ntegratng Equaton (59 wth respect to y, x, w after proper substtuton, we can obtan requred result Equaton (57 Property (3 m+ n+ µ + ν m n µ ν m n µ ν ( F pqrs,,, L w x y f,,, m n µ ν p q r s Proof: By defnton ( of quadruple Laplace transform left hand sde of (5 can be solved as m+ n+ µ + ν m+ n+ µ + ν ( F ( pqrs,,, m n µ ν p q r s m+ n+ µ + ν m+ n+ µ + ν pw qx ry s ( e e e e f(,,, d ddd m n µ ν p q r s m+ n+ µ ν m+ n+ µ + ν pw qx ry s ( e e e e f(,,, d dwxy dd m n µ p q r ν s The nner ntegral wth respect to can be solved by parts and gves Equaton (5 becomes (5 (5 ν s ν ν e f(,,, d ( L{ (,,, } f (5 ν s m+ n+ µ + ν m+ n+ µ + ν ( F ( pqrs,,, m n µ ν p q r s ( { } m+ n+ µ m+ n+ µ + ν pw qx ry ν ν ( e e e ( L (,,, d dd m n f wxy µ p q r ( { } m+ n+ µ m+ n+ µ pw qx ry ν ( e e e L (,,, d dd m n f wxy µ p q r On the smlar lnes, ntegratng Equaton (53 accordngly wth respect to y, x, w, we can obtan requred result Equaton (5 Theorem 3 A functon f( wxyt,,, whch s contnuous on [, and satsfes the growth condton (3 F pqrs,,, as can be recovered from only n3 ( n n+ n+ n3+ n n + n+ + n+ n + n+ n3+ n n n 3 n n n n 3 n f(,,, lm,,, n, n, n3, n n! n! n3! n! w x y n n n3 n w x y To check the effcency of the theorem, we consder the followng example (

7 H U Rehman et al,,, e Let f aw bx cy d for whch Laplace transform can easly be found as F,,, ( pqrs ( p+ a( q+ b( r+ c( s+ d Takng hgher order mxed dervatves of Equaton (5, leads 3 n+ n+ n3+ n n! n! n3! n! ( n+ n+ n3+ n F ( pqrs,,, n+ n+ n3+ n+ n n n n p+ a q+ b r+ c s+ d Usng (5 and theorem 5, yelds (5 n+ n+ n3+ n+ n n n3 n n3 n3 n n n n n n f(,,, lm a+ b+ c+ d+ nn, n3, n w x y w x y n n n n 3 aw bx cy d lm nn, n3, n n n n3 n Usng the applcaton of logarthm and the L Hosptal s rule on the prevous expresson reveals 6 Numercal Examples ln f w, x, y, aw bx cy d, f aw bx cy d,,, e To llustrate the applcablty and effectveness of our method, some examples are constructed n ths secton Example Consder the followng fourth order partal dfferental equaton Applyng quadruple Laplace transform where u(,,, u(,,,, x+ y+ w+ y+ u(, xy,, e, uw (,, y, e, w+ x+ w+ x+ y u( wx,,, e, u( wxy,,, e L on both sdes of Equaton (6, gves ( pqrs U ( p q r s G ( p q r s (55,,,,,,, (6 wxy w wy xy (,,, (,,, + (,,, + (,,, + (,,, wx wy w xy pqf ( p, q,, prf ( p,, r, psf ( p,,, s qrf (, q, r, x y w x qsf (, q,, s rsf (,, r, s + pf ( p,,, + qf (, q,, y + rf (,, r, + sf (,,, s f (,,, ( pqrs ( p ( q ( r ( s G p q r s pqrf p q r pqsf p q s prsf p r s qrsf q r s After substtutng the value, Equaton (6 becomes U,,, ( pqrs ( p ( q ( r ( s Applyng the quadruple nverse Laplace transform on (63 w+ x+ y+ u(,,, L e ( p ( q ( r ( s (6 (63 (6 3378

8 H U Rehman et al Example Consder the followng three-dmensonal dffuson equatons [9] Applyng quadruple Laplace transform where u u u u + +, x y t y+ + 3t + y+ + 3t u( xy,,, x+ y+ e u, yt,, e, u, yt,, e, x+ + 3t x t u x,, t, e, u x,, t, e, x+ y+ 3t x+ y+ + 3t uxy,,, t e, uxy,,, t e, L on both sdes of Equaton (65, gves ( p q r su ( pqrs G( pqrs + +,,,,,,, (65 (,,, yt yt xt xt (,,, + x (,,, + (,,, + y (,,, xyt xyt xy + rf ( pq,,, s + F ( pq,,, s F ( pqr,,, G pqrs pf qrs F qrs qf p rs F p rs ( p + q + r s ( p ( q ( r ( s 3 After substtutng the value of G( pqrs,,,, Equaton (65 becomes U ( pqrs,,, Applyng the quadruple nverse Laplace transform on (67 u( xyt L xyt ( p ( q ( r ( s 3 x+ y+ + 3t,,, e ( p ( q ( r ( s 3 Example 3 Consder the followng non-homogeneous fourth order partal dfferental equaton w x+ y u(,,, + u(,,, 5e, x+ y w+ y u, xy,, e, uw,, y, e, w x w x+ y u wx,,, e, u wxy,,, e (66 (67 (68 Applyng quadruple Laplace transform where L on both sdes of Equaton (69, gves 5 pqrs + U p, q, r, s + G p, q, r, s, ( p ( q+ ( r ( s+ wxy w wy xy (,,, (,,, + (,,, + (,,, + (,,, wx wy w xy pqf ( p, q,, prf ( p,, r, psf ( p,,, s qrf (, q, r, x y w x qsf (, q,, s rsf (,, r, s + pf ( p,,, + qf (, q,, y + rf (,, r, + sf (,,, s f (,,, ( pqrs ( p ( q+ ( r ( s+ G p q r s pqrf p q r pqsf p q s prsf p r s qrsf q r s After substtutng the value of G( pqrs,,,, Equaton (69 becomes (69 (6 3379

9 H U Rehman et al U,,, ( pqrs Applyng the quadruple nverse transform on ( ( p ( q+ ( r ( s+ u L ( p ( q+ ( r ( s+ w x+ y,,, e Example Consder the followng non-homogeneous three-dmensonal dffuson equaton Applyng quadruple Laplace transform where u u u u x y+ t + + e, x y t y t y+ t + + x y+ u( xy,,, e u, yt,, e, u, yt,, e, x+ t x + t u x,, t, e, u x,, t, e, x y x y+ t uxy,,, t e, uxy,,, t e, ( p q r su ( pqrs G( pqrs L on both sdes of Equaton (63, gves + +,,,,,, +, ( p ( q+ ( r ( s+ (,,, yt yt xt xt (,,, + x (,,, + (,,, + y (,,, xyt xyt xy + rf ( pq,,, s + F ( pq,,, s + F ( pqr,,, G pqrs pf qrs F qrs qf p rs F p rs p q r s + + ( p ( q+ ( r ( s+ After substtutng the value G( pqrs,,,, Equaton (63 becomes U ( pqrs Applyng the quadruple nverse transform on (65,,, ( p ( q+ ( r ( s+ x y+ t u( xyt,,, L xyt e ( p ( q+ ( r ( s+ (6 (6 (63 (6 (65 (66 7 Concluson In ths paper, we extend the work of [] [5] to quadruple Laplace transform Exstence and unqueness of the quadruple transform are also dscussed n ths work Some propertes, theorems usng the new quadruple Laplace transform and a table n whch quadruple Laplace transform appled on some functons have also been presented It s analyed that our proposed method s well suted for use n partal dfferental equaton nvolvng four varables Therefore, the present method s an accurate and relable technque for the partal dfferental equatons References [] Kurna, A and Oturanç, G (5 N-Dmensonal Dfferental Transformaton Method for Solvng Partal Dfferental Equatons Internatonal Journal of Computer Mathematcs, 67, [] Evans, DJ, Ergu, M and Bulut, H (3 Varatonal Iteraton Method A Knd of Nonlnear Analytcal Technque: Some Examples Internatonal Journal of Computer Mathematcs, 8,

10 H U Rehman et al [3] Inc, M and Evans, DJ ( An Effcent Approach to Approxmate Solutons of Eghth-Order Boundary-Value Problems Internatonal Journal of Computer Mathematcs, 8, [] Atangana, A (3 A Note on the Trple Laplace Transform and Its Applcatons to Some Knd of Thrd-Order Dfferental Equaton Abstract and Appled Analyss, 3, Artcle ID: [5] Eltayeb, H and Klçman, A (3 A Note on Double Laplace Transform and Telegraphc Equatons Abstract and Appled Analyss, 3, Artcle ID: [6] Dahya, RS and Saber-Nadjaf, J (999 Theorems on N-Dmensonal Laplace Transforms and Ther Applcatons 5th annual Conference of Appled Mathematcs, Unv of Central Oklahoma, Electronc Journal of Dfferental Equatons,, 6-7 [7] Aghl, A and Moghaddam, BS (8 Laplace Transform Pars of n-dmensons and Second Order Lnear Partal Dfferental Equatons wth Constant Coeffcents Annales Mathematcae et Informatcae, 35, 3- [8] Klçman, A and Eltayeb, H ( Some Remarks on the Sumudu and Laplace Transforms and Applcatons to Dfferental Equatons ISRN Appled Mathematcs,, Artcle ID: [9] Chenguel, A and Reghoua, M (3 Ton the Numercal Soluton of Three-Dmensonal Dffuson Equaton wth an Integral Condton Proceedngs of the World Congress on Engneerng and Computer Scence 3, II, 3-5 [] Dtkn, VA and Prudnkov, AP (96 Operatonal Calculus n Two Varables and Its Applcatons Pergaman Press, New York (Englsh Translaton from Russan [] Kanwal, RP (3 Generaled Functons Theory and Applcatons Proceedngs of the World Congress on Engneerng and Computer Scence 3, II,

11 H U Rehman et al Appendx Table of quadruple Laplace transform L for functons of four varables Functon f(,,, Quadruple Laplace transform F ( pqrs,,, abcd abcd pqrs pqrs m n e e Γ µ ν ( + m Γ ( + n Γ ( + µ Γ ( + ν p q r s m+ n+ µ + ν+ aw bx cy d ( p+ a( q+ b( r+ c( s+ d Γ ( + m Γ ( + n Γ ( + µ Γ ( + ν µ ν ( p+ a( m+ ( q+ b( n+ ( r+ c( µ + ( s+ d( ν + m n aw bx cy d cos( aw cos( bx cos( cy cos( d sn ( aw sn ( bx sn ( cy sn ( d cos( w+ x+ y+ sn ( w+ x+ y+ cosh ( aw cosh ( bx cosh ( cy cosh ( d snh ( aw snh ( bx snh ( cy snh ( d pqrs ( p + a ( q + b ( r + c ( s + d abcd ( p + a ( q + b ( r + c ( s + d p + q + r + s pqrs ( p ( q ( r ( s pq + pr + ps + qr + qs + rs ( + p ( + q ( + r ( + s π 6 pqrs pqrs ( p a ( q b ( r c ( s d abcd ( p a ( q b ( r c ( s d ( ew ( fx ( gy ( h aw+ bx+ cy+ d e sn sn sn sn ( efgh ( ( e + p a f + q b g + r c h + s d ( p a( q b( r c( s d ( ew ( fx ( gy ( h aw bx cy d e cos cos cos cos ( ew ( fx ( gy ( h aw+ bx+ cy+ d e snh snh snh snh ( ( ( e + p a f + q b g + r c h + s d ( efgh ( ( p a e q b f r c g s d h ( p a( q b( r c( s d ( ew ( fx ( gy ( h aw bx cy d e cos cos cos cos sn ( aw sn ( bx sn ( cy sn ( d w x y sn ( aw sn ( bx sn ( cy sn ( d cos ( aw cos( bx cos( cy cos( d ( ( ( p a e q b f r c g s d h a b c d tan tan tan tan p q r s 6abcd ( p + a ( q + b ( r + c ( s + d ( p a ( q b ( r c ( s d ( p + a ( q + b ( r + c ( s + d 338

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