A new method for approximate solutions of some nonlinear equations: Residual power series method

Transcription

1 Research Aricle A new mehod for approximae soluions of some nonlinear equaions: Residual power series mehod Advances in Mechanical Engineering 016, Vol. 8(4) 1 7 Ó The Auhor(s) 016 DOI: / aime.sagepub.com Musafa Inc 1, Zeliha S Korpinar, Maysaa Mohamed Al Qurashi 3 and Dumiru Baleanu 4,5 Absrac In his work, a powerful ieraive mehod called residual power series mehod is inroduced o obain approximae soluions of nonlinear ime-dependen generalized Fizhugh Nagumo equaion wih ime-dependen coefficiens and Sharma Tasso Olver equaion subjeced o cerain iniial condiions. The consequences show ha his mehod is efficien and convenien, and can be applied o a large sor of problems. The approximae soluions are compared wih he known exac soluions. Keywords Residual power series mehod, nonlinear ime-dependen generalized Fizhugh Nagumo equaion, Sharma Tasso Olver equaion Dae received: 5 January 016; acceped: 16 March 016 Academic Edior: José Tenreiro Machado Inroducion Parial differenial equaions can define a number of physical problems in differen fields of science. These linear and nonlinear problems play imporan roles in applied science. There are many analyical approximae mehods o solve problems in he lieraure such as he homoopy analysis mehod proposed by Liao, 1, he variaional ieraion mehod proposed by He, 3,4 and homoopy perurbaion mehod. 5,6 Among hese, residual power series mehod (RPSM) is a new algorihm. The RPSM was developed as an efficien mehod for deermining values of coefficiens of he power series soluion for fuzzy differenial equaions. 7 The RPSM is consiued wih a repeaed algorihm. This mehod is effecive and easy o obain power series soluion for srongly linear and nonlinear equaions wihou linearizaion, perurbaion, or discreizaion. Unlike he classical power series mehod, he RPSM does no need o mach he coefficiens of he corresponding erms and a repeaed relaion is no required. This mehod calculaes he coefficiens of he power series by a chain of algebraic equaions of one or more variables. Besides, he RPSM does no require any convering while changing from he higher order o he lower order; hus, he mehod can be applied direcly o he given problem by choosing an appropriae iniial guess approximaion. 1 Deparmen of Mahemaics, Science Faculy, Fira Universiy, Elazığ, Turkey Deparmen of Adminisraion, Faculy of Economic and Adminisraive Sciences, Musx Alparslan Universiy, Musx, Turkey 3 Deparmen of Mahemaics, King Saud Universiy, Riyadh, Saudi Arabia 4 Deparmen of Mahemaics, Cxankaya Universiy, Ankara, Turkey 5 Insiue of Space Sciences, Magurele, Romania Corresponding auhor: Musafa Inc, Deparmen of Mahemaics, Science Faculy, Fira Universiy, Elazığ 3119, Turkey. minc@fira.edu.r Creaive Commons CC-BY: This aricle is disribued under he erms of he Creaive Commons Aribuion 3.0 License (hp:// which permis any use, reproducion and disribuion of he work wihou furher permission provided he original work is aribued as specified on he SAGE and Open Access pages (hps://us.sagepub.com/en-us/nam/ open-access-a-sage).

2 Advances in Mechanical Engineering I has been successfully pu ino pracice o handle he approximae soluion of he generalized Lane Emden equaion, 8 he soluion of composie and noncomposie fracional differenial equaions, 9 predicing and represening he mulipliciy of soluions o boundary value problems of fracional order, 10 consrucing and predicing he soliary paern soluions for nonlinear ime-fracional dispersive parial differenial equaions, 11 he approximae soluion of he nonlinear fracional KdV Burgers equaion, 1 he approximae soluions of fracional populaion diffusion model, 13 and he numerical soluions of linear non-homogeneous parial differenial equaions of fracional order. 14 The proposed mehod is an alernaive process for geing analyic Maclaurin series soluion of problems. This mehod has proved o be powerful and effecive, and can easily handle a wide class of linear and nonlinear problems. The purpose of his work is o employ RPSM o obain he numerical soluion for generalized Fizhugh Nagumo equaion (FNE) wih ime-dependen coefficiens 15 and Sharma Tasso Olver equaion (STOE). 16 Nonlinear ime-dependen generalized FNE is given by 15 u + cos ()u x cos ()u xx + cos ()(u(1 u)(r u)) = 0 (x, ) ½A, BŠ 3 ½0, TŠ, 0 r 1 ð1:1þ subjeced o he iniial condiion u(x, 0)= r + r anh, x ½A, BŠ ð1:þ Using specific soliary wave ansaz and he anh mehod (TanhM), new variey of solion soluions are inroduced in Triki and Wazwaz. 15 Bhrawy 17 applied he Jacobi Gauss Lobao collocaion mehod o solve he generalized FNE. In recen years, many physiciss and mahemaicians have paid much aenion o he FNE on accoun of is imporance in mahemaical physics The following nonlinear equaion is obained u + a u 3 x + 3 a u xx + au xxx = 0 ð1:3þ where a is a real parameer and u(x, ) is he unknown funcion depending on he variable and x. Equaion (1.3) be called STOE in lieraures. The STOE appear in quanum field heory, relaivisic physics, dispersive wave phenomena, plasma physics, nonlinear opics, and applied and physical sciences. 4 8 In addiion, in Jafari e al., 9 fracional sub-equaion mehod is used o consruc exac soluion of he nonlinear fracional STOE. The ouline of he remainder of his aricle is as follows. In secion Numerical applicaions of he RPSM, we presen some properies of RPSM and is numerical applicaions for generalized FNE wih imedependen coefficiens and STOE. Secion Graphical resuls shows formed graphics and drew ables for he reliabiliy of obained soluions. Finally, some concluding remarks are given and graphics are formed in secion Conclusion. Numerical applicaions of he RPSM In his secion, we apply RPSM o solve he aboveproposed equaions. Time-dependen generalized FNE Consider generalized FNE wih ime-dependen coefficiens (1.1) and (1.). The exac soluion for equaion (1.1) is 15 u(x, )= r + r anh r x (3 r) sin () ð Þ We apply he RPSM o find ou series soluion for his equaion subjeced o given iniial condiions by replacing is power series expansion wih is runcaed residual funcion. From his equaion, a repeiion formula for he calculaion of coefficiens is supplied, while coefficiens in power series expansion can be calculaed repeaedly from he runcaed residual funcion. 9,30 Suppose ha he soluion akes he expansion form u = X f n (x) n, 0 \R, x I ð:1þ Nex, we le u k o denoe kh, runcaed series of u k u k = X f n (x) n, 0 \R, x I where u 0 = f 0 (x)=u(x, 0)=f (x): Equaion (.) can be wrien as k ð:þ u k = f (x)+ X f n (x) n, 0 \R, x I, k = 1, n = 1 ð:3þ Firs, o find he value of coefficiens f n (x), n = 1,, 3,..., k in series expansion of equaion (.3), we define residual funcion Res, for equaion (1.1), as Res = u + cos ()u x cos ()u xx + cos ()(u(1 u)(r u)) and he kh residual funcion, Res k, as follows

3 Inc e al. 3 Res k = ðu k Þ + cos () ðu k Þ x cos () ðu k Þ xx + cos ()(u k (1 u k )(r u k )) k = 1,, 3,... ð:4þ As in Abu Arqub and colleagues, 7 10 i is clear ha Res = 0 and lim Res k = Res for each x I and 0: k! Then, ( r Res= r )=0 when = 0 for each r = 0, k. To deermine f 1 (x), we wrie k = 1 in equaion (.4) where for Res 1 = ðu 1 Þ + cos () ðu 1 Þ x cos () ðu 1 + cos ()(u 1 (1 u 1 )(r u 1 )) u 1 = f (x)+f 1 (x) u 0 = f 0 (x)=f (x)=u(x, 0)= r + r Þ xx anh ð:5þ From equaion (.5), we deduce ha Res 1 = 0 ( = 0) and hus f 1 (x)= 1 4 ( 3 + r)r sec h ð:6þ Therefore, he 1s residual power series (RPS) approximae soluions are u 1 = r + r anh ( 3 + r)r sec h ð:7þ Similarly, o find ou he form of he second unknown coefficien, f (x), we wrie u = f (x)+f 1 (x)+ f (x) in Res. ( Res = )=0 ( = 0) and hus f (x)= 1 8 ( 3 + r) r 3 sec h anh ð:8þ Therefore, he nd RPS approximae soluions are u = r + r anh ( 3 + r)r sec h 1 8 ( 3 + r) r 3 sec h anh : ð:9þ Similarly, we wrie u 3 = f (x)+f 1 (x)+ f (x)+ 3 f 3 (x) in Res. ð Res= Þ¼0 ( = 0) and hus f 3 (x)= 1 48 r (3 r + 54r 54r r 4 r cosh () r cosh () 7r cosh () + 7r 3 cosh () 9r 4 cosh () + r 5 cosh ()) sec h 4 ð:10þ Therefore, he 3rd RPS approximae soluions are u 3 = r + r anh ( 3 + r)r sec h 1 8 ( 3 + r) r 3 sec h anh r (3 r + 54r 54r r 4 r cosh () r cosh () STOE 7r cosh ()+7r 3 cosh () 9r 4 cosh ()+r 5 cosh ()) sec h 4 ð:11þ Consider equaion (1.3) wih he iniial condiion 16 u(x, 0)= e x The exac soluion for equaion (1.3) is 16 u(x, )= e (xa) We apply he RPSM o find ou series soluion for his equaion. Suppose ha he soluion akes he expansion form u = X f n (x) n, 0 \R, x I where u k is he runcaed series of u k u k = X f n (x) n, 0 \R, x I ð:1þ ð:13þ where u 0 = f 0 (x)=u(x, 0)=f (x): To find he value of coefficiens f n (x), n = 1,, 3,..., k in series expansion of equaion (.3), we define residual funcion Res, for equaion (1.3), as Res = u + a u 3 x + 3 a u xx + au xxx and he kh residual funcion, Res k, as follows

4 4 Advances in Mechanical Engineering Res k = ðu k Þ + a u 3 k x + 3 a u k xx + aðu k Þ xxx, k = 1,, 3,... ð:14þ To deermine f 1 (x), we wrie k = 1 in equaion (.14) Res 1 = ðu 1 Þ + a u 3 1 x + 3 a u 1 xx + a ð u 1Þ xxx ð:15þ where for u 1 = f (x)+f 1 (x) u 0 = f 0 (x)=f (x)=u(x, 0)= e x From equaion (.15), we deduce ha Res 1 = 0 ( = 0) and hus ex a f 1 (x)= ð1 + e x Þ ð:16þ The 1s RPS approximae soluions are u 1 = e ex a x ð1 + e x Þ ð:17þ Similarly, o find ou he form of he second unknown coefficien, f (x), we wrie u = f (x)+f 1 (x)+ f (x) in Res. ( Res = )=0 ( = 0) and hus f (x)= ex ( 1 + e x )a 1+ ð e x Þ 3 ð:18þ Therefore, he nd RPS approximae soluions are u = e x ex a ð1 + e x Þ ex ( 1 + e x )a 1+ ð e x Þ 3 Similarly, we wrie u 3 = f (x)+f 1 (x)+ f (x)+ 3 f 3 (x) in Res. ( Res = )=0 ( = 0) and hus f 3 (x)= ex (1 4e x + e x )a ð e x Þ 4 ð:19þ Therefore, he 3rd RPS approximae soluions are u 3 = e x ex a ð1 + e x Þ ex ( 1 + e x )a 1+ ð e x Þ 3 and ex (1 4e x + e x )a ð e x Þ 4 3 ð:0þ f 4 (x)= ex ( e x 11e x + e 3x )a 4 4ð1 + e x Þ 5 ð:1þ u 4 = e x ex a ð1 + e x Þ ex ( 1 + e x )a 1+ ð e x Þ 3 ð:þ ex (1 4e x + e x )a ð e x Þ 4 3 ex ( e x 11e x + e 3x )a 4 4ð1 + e x Þ 5 4 Graphical resuls In his secion, we formed graphics and drew ables for he reliabiliy of above-obained soluions. Figures 1 4 show ha he exac error is smaller as he number of k increases. I is clear ha he value of kh runcaed series u k (x, ) affecs he RPS approximae soluions. These figures clearly show ha he convergence of he approximae soluions o he exac soluion relaed o he order of he soluion and he exac error is smaller as he order of he soluion increases. Figure 1. Surface graph of he RPS approximae soluion and exac soluion for equaion (1.1) (r = 0:5).

5 Inc e al. 5 Table 1. Comparison beween RPS approximae soluion u 3 (x, ) and exac soluion of equaion (1.1) (r = 0:5). x : : : : : : : : : : : : : : : : : : : : : : : : : Table. Comparison beween RPS approximae soluion u 4 (x, ) and exac soluion of equaion (1.3) (a = 0:5). x : : : : : : : : : : : : : : : : : : : : : : : : : In Tables 3 and 4, comparison is made among approximae soluions wih known resuls. These resuls are obained using RPSM and TanhM. 15 A comparison is made among approximae soluions wih known resuls. These resuls are obained using RPSM and he modified simple equaion mehod (MSEM). 16 Figure. u k (x, ) soluions of (1.1) equaion when k = 1, 3, 5 versus is exac soluion (r = 0:5, = 1:5). Tables 1 and clarify he convergence of he approximae soluions o he exac soluion. Conclusion The RPSM is applied successfully for solving he generalized FNE wih ime-dependen coefficiens and STOE for cerain iniial condiions. The fundamenal objecive of his aricle is o inroduce an algorihmic form and implemen a new analyical repeaed algorihm derived from he RPS o find numerical soluions for he FNE Figure 3. Surface graph of he RPS approximae soluion and exac soluion for equaion (1.3) (a = 0:5).

6 6 Advances in Mechanical Engineering Declaraion of conflicing ineress The auhor(s) declared no poenial conflics of ineres wih respec o he research, auhorship, and/or publicaion of his aricle. Figure 4. u k (x, ) soluions of (1.3) equaion when k = 1, 3, 7 versus is exac soluion (a = 1:5, = 1). Table 4. Comparison beween soluions u RPSM, u MSEM, and exac soluion of equaion (1.3) (a = 0:5, = 0:5). x u RPSM u MSEM u Exac Table 3. Comparison beween soluions u RPSM, u Tanh, and exac soluion of equaion (1.1) ( = 1, q = 0:). u RPSM u TanhM u Exac and STOE. Graphical and numerical consequences are inroduced o illusrae he soluions. Thus, i is concluded ha he RPSM becomes powerful and efficien in finding numerical soluions for a wide class of nonlinear differenial equaions. The consequences emphasize he power of RPSM in handling a wide variey of nonlinear problems. The RPS does no require linearizaion, perurbaion, or discreizaion of he variables, i is no impressed esimae of errors, and i is no confroned wih necessiy of large calculaor memory and ime. The main advanage of his mehod is he simpliciy in calculaing he coefficiens of erms of he series soluion using only he differenial operaors. Acknowledgemens The auhors would like o express heir sincere graiude o he referees for he valuable suggesions o improve he paper. Funding The auhor(s) disclosed receip of he following financial suppor for he research, auhorship, and/or publicaion of his aricle: This research projec was suppored by a gran from he Research Cener of he Cener for Female Scienific and Medical Colleges, Deanship of Scienific Research, King Saud Universiy. The auhors are hankful for he suppor by he Visiing Professor Program a King Saud Universiy. References 1. Liao SJ. An approximae soluion echnique no depending on small parameers: a special example. In J Nonlinear Mech 1995; 30: Liao SJ. Boundary elemen mehod for general nonlinear differenial operaors. Eng Anal Bound Elem 1997; 0: He JH. Approximae analyical soluion for seepage flow wih fracional derivaives in porous media. Compu Mehod Appl M 1998; 167: He JH. Approximae soluion for nonlinear differenial equaions wih convoluion produc nonlineariies. Compu Mehod Appl M 1998; 167: He JH. Homoopy perurbaion echnique. Compu Mehod Appl M 1999; 178: He JH. A coupling mehod of homoopy echnique and perurbaion echnique for nonlinear problems. In J Nonlinear Mech 000; 35: Abu Arqub O. Series soluion of fuzzy differenial equaions under srongly generalized differeniabiliy. JAdv Res Appl Mah 013; 5: Abu Arqub O, El-Ajou A, Baaineh A, e al. A represenaion of he exac soluion of generalized Lane Emden equaions using a new analyical mehod. Absr Appl Anal 013; 013: Aricle ID El-Ajou A, Abu Arqub O, Al Zhour Z, e al. New resuls on fracional power series: heories and applicaions. Enropy 013; 15: Abu Arqub O, El-Ajou A, Al Zhour Z, e al. Muliple soluions of nonlinear boundary value problems of fracional order: a new analyic ieraive echnique. Enropy 014; 16: Abu Arqub O, El-Ajou A and Momani S. Consrucing and predicing soliary paern soluions for nonlinear ime-fracional dispersive parial differenial equaions. J Compu Phys 015; 93: El-Ajou A, Abu Arqub O and Momani S. Approximae analyical soluion of he nonlinear fracional KdV-Burgers equaion: a new ieraive algorihm. J Compu Phys 015; 93: Alquran M, Al-Khaled K and Chaopadhyay J. Analyical soluions of fracional populaion diffusion model: residual power series. Nonlinear Sud 015; :

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