On computing differential transform of nonlinear non-autonomous functions and its applications

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1 On compung dfferenal ransform of nonlnear non-auonomous funcons and s applcaons Essam. R. El-Zahar, and Abdelhalm Ebad Deparmen of Mahemacs, Faculy of Scences and Humanes, Prnce Saam Bn Abdulazz Unversy, Alhar, 94, KSA. Deparmen of Basc Engneerng Scence, Faculy of Engneerng, Shebn El-Kom, 5, Menofa Unversy, Egyp. Deparmen of Mahemacs, Faculy of Scence, Tabu Unversy, P. O. Box 74, Tabu 749, KSA Absrac: Alhough beng powerful, he dfferenal ransform mehod ye suffers from a drawbac whch s how o compue he dfferenal ransform of nonlnear non-auonomous funcons ha can lm s applcably. In order o overcome hs defec, we nroduce n hs paper a new general formula and s relaed recurrence relaons for compung he dfferenal ransform of any analyc nonlnear non-auonomous funcon wh one or mul-varable. Regardng, he formula n he leraure was found no applcable o deal wh he presen non-auonomous funcons. Accordngly, a generalzaon s presened n hs paper whch reduces o he correspondng formula n he leraure as a specal case. Several es examples for dfferen ypes of nonlnear dfferenal and negro-dfferenal equaons are solved o demonsrae he valdy and applcably of he presen mehod. The obaned resuls declare ha he suggesed approach no only effecve bu also a sragh forward even n solvng dfferenal and negro-dfferenal equaons wh complex nonlneares. Keywords: Dfferenal ransform mehod; Nonlnear non-auonomous funcons; One-dmensonal dfferenal and negro-dfferenal equaons.. Inroducon The Dfferenal Transform Mehod (DTM) whch s based on Taylor seres expanson was frs nroduced by Zhou [] and has been successfully appled o a wde class of nonlnear problems arsng n mahemacal scences and engneerng. The man advanage of he DTM s ha can be appled drecly o nonlnear dfferenal equaons wh no need for lnearzaon, dscrezaon, or perurbaon. Addonally, he DTM does no generae secular erms (nose erms) and does no need o analycal negraons as oher sem-analycal numercal mehods such as HPM, HAM, ADM or VIM and so he DTM s an aracve ool for solvng dfferenal equaons. Alhough hs mehod has been proved o be an effcen ool for handlng nonlnear dfferenal equaons, he nonlnear funcons used n hese sudes are resrced o ceran nds of nonlneares, e.g., polynomals and producs wh dervaves. For oher ypes of nonlnear funcons, Chang and Chang [] consruc a new algorhm based on obanng a dfferenal equaon sasfed by hs nonlnear funcon and hen applyng he DTM o hs obaned dfferenal equaon. Alhough her reamen was found effecve for some forms of nonlneary [,4] sgnfcanly ncreases he compuaonal budge, especally f here are wo or more nonlnear funcons nvolved n he dfferenal equaon beng nvesgaed, as demonsraed by Ebad [5]. Moreover, n he case of complex nonlneares, may be que dffcul o oban he dfferenal equaons sasfed by hese nonlnear funcons. To overcome hs dffculy, a new formula has been derved [5-8] o calculae

2 he dfferenal ransform of any nonlnear auonomous one varable funcon f ( y ). Ths formula has he same mahemacal srucure as he Adoman polynomals bu wh consans nsead of varable componens. We mean by he nonlnear non-auonomous funcons ha hose nonlnear funcons whch conan he ndependen varable and also he dependen varable and/or s dervaves. Unforunaely, for such ypes of complcaed nonlnear non-auonomous funcons wh one or f, y (), =,,.., m, no relaed formula has been gven o calculae her ransform mul-varable ( ) funcons. Ths provdes he movaon for he presen wor. To overcome he drawbac of he DTM ha s how o compue he dfferenal ransform of hgh complex nonlnear non-auonomous funcons, a generalzed formula s deduced n hs paper for compung he dfferenal ransform of any analyc nonlnear non-auonomous funcon wh one or mul-varable. The proposed mehod deals drecly wh he nonlnear non-auonomous funcon n s form whou any specal nds of ransformaons or algebrac manpulaons. Also, here s no need o compue he dfferenal ransform of oher funcons o oban he requred one. For auonomous funcon, as a specal case of he curren sudy, hese formulas and recurrence relaons have he same mahemacal srucure as he Adoman polynomals bu wh consans nsead of varable componens. The effcency of he proposed mehod s dscussed hrough several es examples ncludng nonlnear dfferenal and negro-dfferenal equaons of dfferen ypes.. Dfferenal Transform Mehod The basc defnons and fundamenal heorems of he one-dmensonal DTM and s applcably for varous nds of dfferenal and negro-dfferenal equaons are gven n [, 9-5]. For convenence of he reader, we presen a bref revew of he DTM n hs secon. The dfferenal ransform of a gven analyc funcon y() n a doman D s defned as d y() Y () =! d =, D, () where y() s he orgnal funcon and Y () s he ransformed funcon. The nverse dfferenal ransform of Y () s defned as. () = y = Y ( ) From Eqs. () and (), we ge d y() y() = ( ) =! d. () = From he above proposon, can be found ha he concep of dfferenal ransform s derved from Taylor seres expanson. In acual applcaons, he funcon y() s expressed by a runcaed seres and Eq. () can be wren as N y = Y ( ), (4) = where N s he approxmaon order of he soluon. Some of he fundamenal mahemacal operaons performed by he one dmensonal dfferenal ransform are lsed n Table.

3 Table. Some fundamenal operaons of he one dmensonal dfferenal ransform. Orgnal funcon y() Transformed funcon Y () β v() ± w () β V ( ) ±β W ( ) v() w () = V () W ( ) m d v() ( + m)! m V + m d! m! ( β+ ) m Hm [, ] ( β+ )!( m )! λ! e λ λ e m, where Hm [, ] =,, f m f m < v( τ) dτ V ( ), V ( ) u () v() τ dτ G(), = m d v ( q ) ( + m)! m m q + V + m d! ω π sn( ω +β ) sn( ω +β+ )! ω π cos( ω +β ) cos( ω +β+ )!. Dfferenal Transform formulas If a dfferenal equaon conans an analyc nonlnear non-auonomous funcon f (, y() ) hen he dfferenal ransform F( n) of he funcon (, ()) he followng heorems, where we assume ha f (, y() ) f f ( y() ). f y s requred and can be compued from Theorem. The dfferenal ransform F( n) of any analyc nonlnear non-auonomous funcon f (, y() ) a a pon can be compued from he formula n d F( n) = f, Y () n +λ λ (4) dλ = λ= where Y () s he dfferenal ransform of y(). Proof: The dfferenal ransform F n of (, ()) f y s defned as n d F( n) = f (, y() n ) d =. (5) Subsung Eq. n Eq.5 resuls n

4 n d F( n) = f, Y ( ) n d. (6) = = Now, le =λ, hen Eq.6 becomes n d F( n) = f, Y () n +λ λ. dλ = λ= By hs way, he proof of Theorem s compleed. Theorem. The dfferenal ransform F( n) of any analyc nonlnear non-auonomous funcon f (, y() ) a a pon, sasfes he recurrence relaon where F() f (, Y ()) n F( n) = F( n ) + ( + ) Y F( n ), n =,,.., n = Y () (7) =. Proof: we have f n (, y()) f n (, y() ) f n (, y() ) dy = + y () d (), hen F( n) = f, Y ( ) + f, Y ( ) ( + ) Y ( + ) Y () ( n ) ( n ) = = = = = ( n )! F ( n ) + ( n )! ( + ) Y ( ) F ( n ) = Y () and snce F( n ) s a funcon of and { ()} n Y, hen = n F( n) = F( n ) + ( + ) Y F( n ) n = Y () By hs way, he proof of Theorem s compleed. Thus by Theorems and we have mplemened a new algorhm for compung he one-dmensonal dfferenal ransform of any analyc nonlnear funcon non-auonomous f y. (, ()) We observe ha for auonomous funcon, he defnon (4) and recurrence relaon (7) are reduced o, respecvely n d F( n) = f Y () n λ, (8) dλ = λ= n F( n) = ( + ) Y ( ) F( n ), n =,,... (9) n = Y ( ) One can observe ha he resul n (8) s he presen formula n [5-8] 4

5 In fac f a sysem of one-dmensonal dfferenal equaons conans a coupled analyc f, y (),,,.., m f, y () f f y () f y ()... f y () nonlneary ( ) =, where we assume ha m m, hen Theorems and can be easly exended usng he above presen procedure o mul-varable funcon and sasfy he followng recurrence algorhms Corollary. The dfferenal ransform F n of any analyc non-auonomous funcon (, ()) f y, =,,.., m a a pon can be defned by n d F( n) = f, Y () n +λ λ dλ, =,,.., m () where Y () s he dfferenal ransform of y (). = λ= Corollary. The dfferenal ransform =,,.., m a a pon, sasfes he recurrence relaon F n of any analyc non-auonomous funcon (, ()) f y, m n F( n) = F( n ) + ( + ) Y ( ) F( n ), n =,,.., n = = Y ( ) () where F() f (, Y ()) =. Moreover for auonomous funcon, he defnon () and recurrence relaon () are reduced, respecvely, o n d F( n) = f Y () n λ, =,,.., m, () dλ = λ= m n F( n) = ( + ) Y ( ) F( n ), =,,.., m, n =,,.., n () = = Y ( ) whch have he same mahemacal srucure as he Adoman polynomals [6] bu wh consans nsead of varable componens. 4. Resuls In hs secon, we have solved dfferen ypes of dfferenal and negro-dfferenal problems wh dfferen forms of nonlnear non-auonomous funcons o demonsrae he valdy and applcably of he presen mehod. Example. Consder he nonlnear nal-value problem y () y() = ln + y(), () y =. (۱٤) Usng he basc properes of DTM and ang he ransform of equaons n (4) resul n 5

6 ( + ) Y ( + ) Y ( ) = F( ), Y () =, =,,,..., (5) where F( ) s he dfferenal ransform of he nonlnear erm ln( + y). F( ) s compued usng he presen mehod and gven by ( Y ), Y () ( + Y () ), Y ( + Y ()) ( )) ( Y )) 4 Y ( Y )) 4( Y ()) + Y () F() = ln( + Y ()), F() = F() = + Y () + () Y () Y () + Y () + Y ( F () = +, + Y () + () + ( Y (4) Y () + Y () Y () + Y () () + Y () F (4) = + + Y () + Y () + Y () + ( + 4 Therefore, a combnaon of (5) and (6) resuls n he seres soluon. (6) y = For suffcenly large number of erms, he closed form of he soluon s y() e =, whch s he exac soluon. Table shows he absolue error obaned for hree varous numbers of erms and a some es pons. Table : Numercal resuls for Example. y y N = 5 N = N = e-8 6.e e e-6.869e- 4.6e e e- 5.55e e-4.48e e-5..65e-.7e-8 5.8e-4 Example. Consder he nonlnear nal-value problem y () + εy() = εsn( y()), y () =. (7) Tang he dfferenal ransform of equaons n (7) resuls n ( + ) Y ( + ) + ε Y Y ( ) Y ( ) = εf( ), Y () =, =,,,..., (8) = where F( ) s he dfferenal ransform of he nonlnear erm sn( y ( )). F( ) usng he presen mehod and gven by s compued 6

7 Y () F() =, F() = Y (), F() = Y (), F() = + Y (), 6 5 F(4) = y() Y () y(), F(5) = Y Y () Y () Y () Y + Y (4) Usng (8) and (9), he seres soluon s obaned and gven a ε =. by. (9) y = The presened resuls are compared wh hose obaned usng MATLAB bul-n solver ode45 n Table. The ode45 solver negraes ODEs usng explc 4h & 5h Runge-Kua (4, 5) formula [7]. In order o guaranee a good numercal reference, ode45 s confgured usng an absolue error of 8 and relave error of. Table : Numercal resuls for Example. y( ) y ( ) N = 5 N = N = e-6 8.e- 9.66e e-4.79e-6.9e e-.5e-4.e e-.4e-.654e e-.556e- 8.8e- Example. Consder he followng nonlnear problem wh mulple soluons ( ) y () y() + = +, () y =. () The dfferenal ransform of equaons n () are = + () ( + ) Y ( + ) H(, ) = F( ) ( )! ( + )! and Y () =, where F( ) s he dfferenal ransform of he nonlnear erm + y(). Applyng he presen mehod o he nonlnear funcon f (, y()) = + y() a = and Y () =,resuls n Y () F() =±, F() =±, F() =± +, F() =± Y () + Y () Y () F(4) =± Y () + Y () + ( 8 Y () 64 Y ()) Y () + Y (), () 4 Y () Y () Y () 5 Y () Y () Y () 7 Y () F(5) =± + Y (4) + Y () + + Y () Y () where he braces wh posve sgn are due o usng he prncpal square roos whle he braces wh negave sgn are due o usng he negave roos. Therefore, a combnaon of () and () resuls n wo seres soluon gven by 7

8 y () =±, whch are he exac soluons of (). Example 4. Consder he nonlnear frs order Volerra negro-dfferenal equaon y = cos + + sn ( + y) d τ τ τ, y =, y =. () The dfferenal ransform of equaons n () are π δ( ) F( ) ( + ) Y ( + ) = cos +δ ( ) +, Y () =, Y () =,!, where F( ) s he dfferenal ransform of he nonlnear erm sn ( τ + y ( τ )). By applyng he presen mehod o he nonlnear funcon f ( τ, y( τ)) = sn ( τ + y( τ )) a τ =, Y () = and Y () =, and by usng he prncpal value of he square roo and nverse rgonomerc funcons, we ge F() =, F() =, F() = Y (), F() = Y () +, F(4) = Y (4) + Y (), 6 F(5) Y (5) Y () Y (), F(6) Y () Y (4) Y () Y () Y (6) Y () = = F(7) 56 Y () 84 Y () Y () Y (4) Y () Y (5) Y (7) Y () = Hence, he seres soluon s obaned and gven by. () y = For suffcenly large number of erms, he closed form of he soluon s y = sn +, whch s he exac soluon. Table 4 shows he absolue error obaned for hree varous numbers of erms and a some es pons. Table 4: Numercal resuls for Example 4. y( ) y ( ) N = 5 N = N = e e-6.e e-7.497e- 5.55e e e-.e e-5.4e-9.e e-4.489e e-5 Example 5. Consder he nonlnear second order Volerra negro-dfferenal equaon 8

9 sec τ y () y() y () = + dτ + y ( τ ), y () =, y () =. (4) The dfferenal ransform of equaons n (4) are F( ) ( + )( + ) Y ( + ) = ( + ) Y ( + ) Y ( ) δ( ) +, Y () =, Y () =, Y () =, = sec τ,where F( ) s he dfferenal ransform of he nonlnear erm obaned usng he presen + y ( τ ) mehod a τ = and gven by F() =, F() =, F() =, F() =, F(4) = Y (), Y (5) = Y (4), F(6) = Y (5) + Y () Y (), F(6) = ( Y ()) Y (4) Y (6), 45 7 F(7) 5 Y () Y () Y () Y (5) Y (5) Y (7) Y (4) = Hence, he approxmae seres soluon s obaned and gven by (5) 7 6 y = For suffcenly large number of erms, he closed form of he soluon s y = an, whch s he exac soluon. Table 5 shows he absolue error obaned for hree varous numbers of erms and a some es pons. Table 5: Numercal resuls for Example 5. y( ) y ( ) N = 5 N = N = e-7.845e- 8.49e e-5.975e-7.89e e-.7649e-5.69e e-.8e-.799e e-.49e- 9.9e-4 Example 6. Consder he nonlnear Volerra negro-dfferenal equaon wh proporonal delay ( τ ) y sn y = sn + d τ (τ + ) The dfferenal ransform of equaons n (6) are, y () =, y () =. (6) + F π π ( ) ( + ) Y ( + ) = δ( ) sn δ( ) + sn, Y () =, Y () =,,!! = = 9

10 where F( ) s he dfferenal ransform of he nonlnear erm w τ, w ( τ) = Y ( ) τ (τ + ) obaned usng he presen mehod and gven by = F() =, F() =, F() = 8 Y (), F() = 54 Y () + 54 Y (), F(4) = 8 Y () + 6 Y () 6 Y () + 6 Y (4), F(5) = 486 Y () + (486 Y () 486) Y () 486 Y (4) Y (5) Y () Hence, he seres soluon s obaned and gven by. (7) y () = +, whch s he exac soluon. Example 7. Consder he followng nonlnear non-auonomous nal-value ODE sysem y () = y() + + ln y() + y () 4 y () = y () + ln ( + y () ). (8) y() y() =, y () = Applyng he dfferenal ransform o (8), resuls n + + =δ + ( ) Y ( ) ( K ) Y ( ) F( ) ( + ) Y ( + ) = Y ( ) δ ( K) + F ( ). (9) Y () =, Y () = where F( ) and F are he dfferenal ransform of he nonlnear funcons 4 f = ln y and f = ln ( + y ) + y () y(), respecvely. F ( ) and F ( are compued usng formula() and gven by F() =, F() = Y () + + Y (), F() = Y () ( + Y () ) + Y () ( Y () + + Y () ) F() = Y () + ( + Y () ) ( + Y () ) Y () + Y (), ( Y () Y () Y () + Y () )( Y () + + Y () ) + ( Y () + + Y () ) () F() =, F() = Y () Y () F() = Y () Y () Y () + ( + Y () ), F() = Y () + Y () Y () Y () Y () + ( + Y () ) Y () ( + Y () ) 4 () Hence, he seres soluons are obaned and gven as y =

11 y = For suffcenly large number of erms, he closed forms of he soluons are () y = e, y () = e, whch are he exac soluons. Table 6 shows he absolue error obaned for hree varous numbers of erms and a some es pons. Table 6: Numercal resuls for Example 7 y y y y N = 5 N = N = 5 N = 5 N = N = e-7.e-5.5e e e-6.4e e-5.e e-6 6.e-6.87e-.4e e-4.79e-.4e-6 7.8e e- ٤ ٥۱۰۱e e-4 4.7e-9.555e-5 4.6e-4.48e e e- 4.69e-8 9.9e-4.65e-.7e e-4 Tables 5 show he absolue errors of he presen mehod for dfferen approxmaon order, N, of he soluon and a some es pons. The resuls show ha he obaned soluons are accurae and converge o he exac ones wh ncreasng he order N. 5. Conclusons In hs paper, a new general formula and hence new recurrence relaons have been derved for compung he dfferenal ransform of any analyc nonlnear non-auonomous funcons wh one or mul-varable. As a specal case of he presen sudy,.e., for auonomous funcon, he curren formulas and recurrence relaons reduces o he same mahemacal srucure as he Adoman polynomals bu wh consans nsead of varable componens n he leraure. I was found ha, he suggesed mehod deals drecly wh he nonlnear non-auonomous funcons, where specal ransformaon or algebrac manpulaons were compleely avoded. In addon, he mehod has been successfully appled on dfferen ypes of dfferenal and nego-dfferenal equaons. Moreover, he obaned seres soluons demonsrae he valdy and applcably of he presen approach. Numercally, he resuls showed ha a fas convergence has been acheved for he obaned seres soluons. An advanage of he presen mehod s ha can be combned wh Padé Approxman (PA), Aferreamen Technques (AT), Power Seres Exender Mehod (PSEM) and mul-sep echnque, among many ohers. Fnally, he auhors beleve ha he presen sudy should be exended o nclude smlar dfferenal and nego-dfferenal equaons n he appled scences, whch ncreases s applcably. Conflcs of Ineres: Auhor has declared ha no compeng neress exs. References. Zhou, J. K., 986. Dfferenal Transformaon and s Applcaons for Elecrcal Crcus, Huazhong Unversy Press, Wuhan Chna 986. Chang, S. H., & Chang, I. L. (8). A new algorhm for calculang one-dmensonal dfferenal ransform of nonlnear funcons. Appled Mahemacs and Compuaon, 95(),

12 . Ebad, A. (). Approxmae perodc soluons for he non-lnear relavsc harmonc oscllaor va dfferenal ransformaon mehod. Communcaons n Nonlnear Scence and Numercal Smulaon, 5(7), Ebad, A. (). A relable aferreamen for mprovng he dfferenal ransformaon mehod and s applcaon o nonlnear oscllaors wh fraconal nonlneares. Communcaons n Nonlnear Scence and Numercal Smulaon, 6(), Ebad, A. (). On a new dfferenal ransformaon mehod for solvng nonlnear dfferenal equaons. Asan-European Journal of Mahemacs, 6(4), Behry, S. H. (). Dfferenal Transform Mehod for Nonlnear Inal-Value Problems by Adoman Polynomals. Journal of Appled & Compuaonal Mahemacs,. 7. N, H. S., & Soleyman, F. (). A Taylor-ype numercal mehod for solvng nonlnear ordnary dfferenal equaons. Alexandra Engneerng Journal, 5(), Faoorehch, H., & Abolghasem, H. (). Improvng he dfferenal ransform mehod: a novel echnque o oban he dfferenal ransforms of nonlneares by he Adoman polynomals. Appled Mahemacal Modellng, 7(8), Odba, Z. M. (8). Dfferenal ransform mehod for solvng Volerra negral equaon wh separable ernels. Mahemacal and Compuer Modellng, 48(7), Abdulaw, M. (5). Soluon of Cauchy ype sngular negral equaons of he frs nd by usng dfferenal ransform mehod. Appled Mahemacal Modellng, 9(8), EL-Zahar, E. R. (). Approxmae analycal soluons of sngularly perurbed fourh order boundary value problems usng dfferenal ransform mehod, Journal of Kng Saud Unversy (Scence), 5(), El-Zahar, E. R. (5). Applcaons of Adapve Mul sep Dfferenal Transform Mehod o Sngular Perurbaon Problems Arsng n Scence and Engneerng, Appled Mahemacs and Informaon Scences, 9(), -.. El-Zahar, E. R. (6). Pecewse approxmae analycal soluons of hgh order sngular perurbaon problems wh a dsconnuous source erm, Inernaonal Journal of Dfferenal Equaons, vol. 6, Arcle ID 564, pages, 6. do:.55/6/ Erür, V. S., Odba, Z. M., & Moman, S. (). The mul-sep dfferenal ransform mehod and s applcaon o deermne he soluons of non-lnear oscllaors. Advances n Appled Mahemacs and Mechancs, 4(4), Aroglu, A., & Ozol, I. (8). Soluons of negral and negro-dfferenal equaon sysems by usng dfferenal ransform mehod. Compuers & Mahemacs wh Applcaons, 56(9), Duan, J. S. (). Convenen analyc recurrence algorhms for he Adoman polynomals. Appled Mahemacs and Compuaon, 7(), Dormand, J. R., & Prnce, P. J. (98). A famly of embedded Runge-Kua formulae. Journal of compuaonal and appled mahemacs, 6(), 9-6.