Mathematical Theory and Modeling                                                                www.iiste.orgISSN 2224-580...
Mathematical Theory and Modeling                                                                            www.iiste.orgI...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                  www.iiste.orgISSN 2224-5...
Mathematical Theory and Modeling                                                         www.iiste.orgISSN 2224-5804 (Pape...
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Solution of a singular class of boundary value problems by variation iteration method


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Solution of a singular class of boundary value problems by variation iteration method

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011 Solution of a Singular Class of Boundary Value Problems by Variation Iteration Method Bhupesh K. Tripathi Department of Mathematics, C.M.P. College, University of Allahabad, Allahabad-211002, India Email: bupeshkt@gmail.comAbstractIn this paper, an effective methodology for finding solution to a general class of singular second orderlinear as well as nonlinear boundary value problems is proposed. These types of problems commonly occurin physical problems. The solution is developed by constructing a sequence of correctional functional viavariation Iteration theory. The analytical convergence of such occurring sequences befitting to the contextof the class of such existing problems is also discussed. The efficacy of the proposed method is tested onvarious problems. It is also observed that execution of only few successive iterations of correctionfunctionals may lead to a solution that is either exact solution or very close to the exact solution.Keywords: Variation iteration method, sequence, linearization, discretization, transformation Convergence,Lagrange multiplier, smooth function, B-Spline, projection method, Lie group1. IntroductionA wide spectrum of well defined properties and behavior systematically associated to a class ofevents/situations occurring on varied fronts in celestial bodies or multidisciplinary sciences either internallyor externally or both ways simultaneously are realized or discerned in real or abstract sense. When theseproblems are modeled mathematically in order to envisage or acknowledge the endowed and all inherentcharacteristics in and around thereof, a class of second order singular differential equations along with twoboundary conditions comes into coherent consideration. Therefore, for such class a suitable and sustainablesolution either numerically appropriate or analytically in the exact form, is must and equally important inwhatsoever manner it is made possible by applying so any feasible proposed variant.Consider a general class of boundary value problems as follows x −α (x α y / )/ = f (x, y) 0< ≤ 1 (1.1) y(0) =A , y(1) =BA, B are constants andα ∈ ℝ − set of real numbers. The function f(x, y) is a real valued continuous ∂ffunction of two variables x and y such that (x, y) ∈ ℝ × ℝ and that is a nonnegative and continuous ∂yfunction in a domain R = {(x, y) :(x, y)∈[0 1]× ℝ}. Solution to such class of problems exists [7-8]. Out ofsuch class it plausible to consider a sub-class formed when α ∈(0 1)⊆ ℝ for elaborated analysis anddiscussion of facts. The class of problems (1.1) from a specific area of the field of differential equation hasbeen a matter of immense research and keen interest to researchers in recent past. Several methods likeB-Spline, homotopy method, Lie group analysis, power series method, projection method, Adomianmethod, multi- integral method, finite difference method [9- 15] have been applied on to justify animmaculate importance of such class of problems. Variation iteration method, a modified Lagrange method[16] originally proposed by He [17-21], stands recognized as promising and profusely used method ofresearch in almost all disciplines of science and technology as an alternative method which is different fromother methods of linearization, transformation and discretization used to solve such type of problems insome way or other way round. It is pertinent to note that the proposed method has fared well, over a largeclass of mathematically modeled problems whenever or wheresoever’s such a suitable situation has havearoused and it is demanded to be applied so. Eventually, credit accrue to variation iteration method forsolving a class of distinguished and challenging problems like, nonlinear coagulation problem with mass 1
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011loss ,nonlinear fluid flow in pipe-like domain, nonlinear heat transfer, an approximate solution for onedimensional weakly nonlinear oscillations, nonlinear relaxation phenomenon in polycrystalline solids,nonlinear thermo elasticity, cubic nonlinear Schrodinger equation, semi-linear inverse parabolic equation,ion acoustic plasma wave, nonlinear oscillators with discontinuities ,non-Newtonian flows, Burger’s andcoupled Burger’s equation, multispecies Lotaka –Volterra equations, rational solution of Toda latticeequation, Helmholtz equation, generalized KdV equation[17-34].2. Variation Iteration Method (VIM)The basic virtue and fundamental principle associated to variation iteration method may be expressed inbrief by considering a general differential equation involving a differential operator D as follows. Let Dy(x) = g(x) x∈Ι⊆ℝ (2.1)y(x) is sufficiently smooth function on some domain Ω and g(x) an inhomogeneous real valued function.(2.1) can be rewritten as, L (y(x)) + N (y(x)) = g(x) x∈Ι⊆ℝ (2.2)where L and N are linear and nonlinear differential operators, respectively.Ostensibly, the privileged variation iteration method has natural aptness and basic tendency to generate arecursive sequence of correction functionals that commands and allows to conserve a real power andabsolute potential for finding a just and acceptable solution to the given class of problems (1.1) and thesequence of correctional functional over(2.2) is x ̃ yn+1 (x) = yn (x) + ∫ μ(s) ((L (yn (s)) +N (yn (s)) – g (s)) ds , n≥0 (2.3) 0where μ stands for Lagrange multiplier determined optimally satisfying all stationary conditions after thevariation method is applied to (2.3). The importance and therefore utility of method all over lies with theassumption and choice of considering the concerned inconvenient highly nonlinear and complicateddependent variables as restricted variables thereby minimizing its magnitude, the accruing error that mighthave crept into the error prone process while finding a solution to (1.1). As aforementioned, yn is the ̃restricted variation, which means δyn ̃=0. Eventually, after desired μ is determined, a proper and suitableselective function (linear or nonlinear) with respect to (2.2) is assumed as an initial approximation forfinding next successive iterative function by recursive sequence of correction functional. Thereafterboundary conditions are imposed on the final or preferably on limiting value (as n → ∞) of sequentialapproximations incurred after due process of iteration.3. Variational Method and Lagrange MultiplierThe variational method and Lagrange multiplier are convoluted corresponding to (1.1) by the iterative andsuccessive correction functional relation as x / yn+1 (x) ̃ = yn (x) + ∫ μ(s) (s α yn (s))/ -x α f(s, yn (s))) ds n≥0 (3.1) 0where yn (x) is nth approximated iterative solution of (1.1). Suppose optimal value of μ(s) is identifiednaturally by taking variation with respect to yn (x) and subject to restricted variation δyn =0. Then from ̃(x)(3.1) we have x / ̃ δyn+1 (x) =δyn (x) +δ ∫ μ(s)((s α yn )/ -s α f(s, yn (s)) ds n≥0 (3.2) 0Integrating by parts and considering the restricted variation of yn (i.e. δyn =0) as well relation (3.2) gives / x δyn (x) = (1- μ/ (s)) δyn (x) + δ(μ(s) s α yn (s)) |s=x + ∫ (μ/ (s) s α )/ δyn (s)ds, 0 n ≥ 0Therefore, the stationary conditions are μ/ (s) s α = 0, μ(x) = 0 , (μ/ (s)s α )/ = 0It gives S1−α −X1−α µ(s)= 1−αFrom (3.1), the sequence of correction functionals is given by 1 x yn+1 (x) =yn (x) + ∫ (s α -x α ) ((s α yn (s))/ -s α̃yn (s))ds f(s, n≥ 0 (3.4) 1−α 0 2
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011 ∞It may be deduced from (3.4) that the limit of the convergent iterative sequence {yn } , if it converges n=1on satisfying given boundary conditions, is the exact solution to (1.1).4. Convergence of Iterative SequenceIn order to carry out convergence analysis of the sequence of correctional functionals generated byexecution of VIM with respect to given class (1.1) in view of (3.1), we consider yn+1(x) = yn (x) +∑n−1(yk+1 (x) − yk (x)) is the nth partial sum of the infinite series k=0 y0 (x) +∑∞ (yk+1 (x)−yk (x)) k=0 (4.1)And that convergence of auxiliary series (4.1) necessarily implies the convergence of iterative sequence {yn (x)}∞ of partial n=1 sums of the series (4.1).Let y0 (x) be the assumed initial selective function. The first successive variation iterate is given by x / y1 (x)=∫ μ (s)((s α y0 (s))/ _s α f(s, y0 (s)))ds 0 (4.2)Integrating by parts and in sequel applying the existing stationary conditions, we have x / |y1 (x)− y0 (x)|=|∫ (y0 (s)+μ(s)s α f(s, y0 (s))ds| 0 (4.3) x Or |y1 (x)-y0 (x)|≤ 0 | y0 (s)|+|s α ||µ ∫( 1 (s)||f(s,y0 (s)|)ds x 1 Or | y1 (x)-y0 (x) | ≤ ∫ ( | y0 (s) | + |µ || f(s,y0 (s)|) 0 (s) ds (4.4)Again pursuing similar steps as in (4.2) and adopting usual stationary conditions likewise, relation (3.4)gives x |y2 (x) − y1 (x)|=|∫ μ(s) s α(f(s),y1 (s))−f(s,y0 (s))ds| 0 (4.5) x Or, |y2 (x) − y1 (x)|≤ ∫ |μ(s)||s α |(f(s),y1 (s))−f(s,y0 (s))|ds 0 x Or, |y2 (x) − y1 (x)|≤ ∫ |μ(s)| (f(s), y1 (s))− f(s, y0 (s)) 0 |ds (4.6)In general, we have x |yn+1 (x)−yn (x)|=|∫ μ(s)s α (f(s,yn (s))−f(s,yn−1 (s)))ds| 0 (4.7) x Or, |yn+1(x) – yn (x)|≤ ∫ |μ(s) ||s α || (f(s,yn (s)) −f(s,yn−1 (s))) 0 |ds ∀ n ≥ 2 x Or, |yn+1 (x) −yn (x)|≤ ∫ |μ(s) || (f(s,yn (s)) −f(s,yn−1 (s))) |ds 0 ∀ n ≥2 (4.8) ∂f(x,y)Since f(x, y) and are continuous on R, therefore for fix sϵ [0 1] and by virtue of ∂ymean value theorem ∃ (s,θ0 (s))∈ R satisfying (say, yn−1 (s) θ0 (s) yn (s)), n n∀n ∈ IN , s≤ x ≤ 1 , such that ∂f(s,θ0 (s)) n+1 |f(s, yn (s)) −f(s, yn−1 (s))| = | ||yn (s) −yn−1 (s)| ∀ n ≥ 2 (4.9) ∂yNow, suppose 1 / M∞ =sup (|y0 (s) |+|μ(s)|| f(s, y0 (s))| , s ≤ x ≤ 1 (4.10) ∞ ∂f(s,θ0 (s)) n and M2 =sup (|μ(s)|| |) , s ≤ x ≤ 1 , n∈ IN (4.11) ∂yAgain to begin with assume 1 2 M=sup(M∞ ,M∞ ) (4.12)We observe and proceed to establish the truthfulness of the inequality Mn+1 xn+1 |yn+1 (s) −yn (s)| ≤ ∀n ∈ IN (4.13) n+1!Relations (4.4), (4.10), (4.9) and (4.12) give x x | y1 (x)-y0 (x) | ≤ ∫ M1 ds o ≤ ∫ M ds ds 0 = Mx (4.14) ∂f(s,θ0 (s)) xAs well as, |y2 (x) − y1 (x)|≤ sup|μ(s)|| 1 |∫ |(y1 ( 0 s)) − y0 (s) )|ds ∂f(s,θ0 (s)) ∂y x 1 x M2 x2 or |y2 (x) − y1 (x)|≤ sup(|μ(s)|| | ∫ |(y1 ( 0 s)) − y0 (s))|ds=M∫ M ds= 0 ∂y 2 s ≤ x ≤ 1 ,n∈ IN 3
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011Thus, the statement (4.13) is true for natural number n=1 Mn xnSuppose that |yn (s) −yn−1 (s)| ≤ holds for some, n ∈ IN n!Then, relations (4.8), (4.9) and (4.12) imply x ∂f(s,θ0 (s)) n |yn+1 (x)−yn (x)|≤ ∫ |μ(s)|| 0 ||yn (s)−yn−1 (s)|ds ∂y x ∂f(s,θ0 (s)) n i.e. |yn+1 (x) - yn (x ) | ≤ ∫ |sup(μ(s))| (sup| ∂y 0 |)|yn (s) −yn−1 (s) |ds s≤x≤1 n ∈ IN ∂f(s,θ0 (s)) n+1 xor , |yn+1(x) −yn (x)| ≤ sup(|μ(s)| | | ∫0 |yn (s) − yn−1 (s)|ds ∂y x Mn sn Mn+1 xn+1 ≤ M∫0 ds= n! n+1!Therefore, by Principle of Induction Mn+1 xn+1|yn+1 (x) − yn (x) |≤ holds ∀xϵ [0 1] and ∀n ∈ IN. n+1!So the series (4.1) converges both absolutely and uniformly for all x ∈ [0 1] Mn+1 xn+1Since, |y0 (x)|+∑∞ |yn+1 (x) −yn (x)|≤ |y0 (x)|+∑∞ n=0 n=0 =| y0 (x)|+ (eMx −1), ∀x ∈[01] n+1!Asserting that the series y0 (x) +∑∞ (yk+1 (x)−yk (x)) converges uniformly ∀x ∈ [01] and hence the k=0sequence of its partial sums {yn (x)}∞ converges to a limit function as the solution. n=05. Numerical ProblemTo begin with implementation and analyze scope of VIM, we apply this very method to find the solution oflinear and nonlinear problems that have been solved by different methods in literature. Specifically tomention is the method to solve it numerically and via numerical finite difference technique of solution.Example 1: Consider the following boundary value problem [12] α y (2) (x)+ y (1) (x) = −x1−α cos x −(2−α)x1−α sin x (5.1) x y(0) = 0 , y(1) = cos 1Solution: To solve this we construct correction functional as follows x / yn+1 (x) = yn (x) + ∫ μ(s) ((−s α yn (s))/ − s cos s – (2−α) sin s) ds , 0 n≥0where μ(s) Is optimally identified Lagrange multiplier similar to (3.3). The first iterative solution isgiven by x / y1 (x) = yo (x) + ∫ μ(s) ((−s α y0 (s))/ − s cos s – (2−α) sin s) ds 0 Since the selective function y0 (x) is arbitrary for simplicity and easiness we may choose / y0 (x) = a 0 x1−α , so that (−s α y0 )/ ) = 0 x Thus, y1 (x) = a 0 x1−α + ∫ μ(s) 0 (−s cos s – (2−α) sin s) ds Now performing usual simplifications and applying term by term series integration, we get x2n+3 x2n+1−α y1 (x) = a 0 x1−α −[ ∑∞(−1)n 0 (2n+3−α)(2n+1)! + (1−α) ∑∞ (−1)n+1 n=1 (2n+1−α)(2n)! ] 1−α x2n 1−α or y1 (x) = a 0 x + x ∑∞ (−1)n n=1 (2n)! 2n x or y1 (x) = a 0 x1−α +x1−α ∑∞ (−1)n n=0 − x1−α (2n)! i.e. y1 (x) == (a 0 − 1) x1−α + x1−α cos x (5.2)In order to match the boundary condition y(1) = cos(1) taking limit as (x → 1) we find a0 = 1 ,only the first iterate giving the exact solution as y(x)= y1 (x) = x1−α cos xExample-2: Consider the boundary value problem [12] (x α y / )/ = βx α+β−2 ((α + β − 1) + βx β ) y 4
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011 y (0) =1 , y(1) =exp (1) (5.3)Solution: The correction functional for the problem (5.3) is x / yn+1 (x) =yn (x) + ∫ μ(s) ((s α yn )/ − β (α + β − 1)s α+β−2 − β2 s α+β−2 ) yn (s) 0 (5.4)μ(s) Is optimally identified Lagrange multiplier similar as (3.2)Inserting, y (0)=y0 (x)=1 to (5.4) when n=1, as selective initial approximation function we process outfollowing induced successive iterative approximate solutions as x2β y1 (x) = 1+x β +β 2(α+β−1) x2β x3β y2 (x) = 1+x β + +β 2.1 3(α+3β−1) x2β x3β x4β y3 (x) = 1+x β + + +β 2.1 3.2.1 4.2(α+4β−1) x2β x3β x4β x5β y4 (x) = 1+x β + + + +β 2.1 3.2.1 5.3.2(α+5β−1) x2β x3β x4β x5β x6β y5 (x) = 1+x β + + + + +β 2.1 3.2.1α+6β−1)Similarly, continuing in like manner inductively we find the general term of the sequence x2β x3β x4β x5β x6β xnβ nβx(n+1)β yn (x) = 1+x β + + + + + +………… + + 2.1 3.2.1 n! n+1!(α+(n+1)β−1) xkβ nβx(n+1)β i.e. yn (x) = ∑n k=0 + (5.5) k! n+1!(α+(n+1)β−1) nβx(n+1)βNow, we observe that Tn = (say), is the general term of a convergent n+1!(α+(n+1)β−1) nβx(n+1)βSeries ∑∞ n=0 . n+1!(α+(n+1)β−1) nβx(n+1)βTherefore, lim (n→ ∞) = 0 and (5.5) facilitates the exact solution to (5.3) as x kβ n+1!(α+(n+1)β−1)y(x) =lim (n→ ∞)( ∑n k=0 k! ) =exp (x β ).Example-3: Consider the boundary value problem [9] βxα (x α y / )/ = (βx β ey − (α + β − 1)) 4+xβ 1 1 y (0) =ln , y(1) = ln (5.6) 4 5 1Solution: Let, y0 = y(0) = ln , be the selective initial approximation function .Then by VIM 4First iterative approximate solution to (5.6) simplifies to 1 x μ(s) β2 sα+2β−2 y1 (x) = ln + ∫0 ( − (α+β − 1) βα s α+β−2 ) ds (5.7) 4 4+xβ 4where as μ(s) is optimally identified Lagrange multiplier as existing in (3.3) and after simplifying (5.4) 1the required first approximate solution to (5.6) satisfying the given boundary condition y(0) = ln is 4as follows 1 xβ 1 xβ α+2β−1 (−1)n xβ y1 (x) =ln − + ( )2 +∑∞ ( n=3 )( ) ( )n (5.8) 4 4 2 4 α+nβ−1 n 4Now, we observe in (5.8) that the first three terms of the first approximate iterative solution of (5.6) matchthe first three terms of the expanded Taylor’s series solution even though only first boundary condition is 5
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.4, 2011being used so for. However, if we allow β to tend to zero in (5.8) βas β is arbitrary, y1 (x) violates the 1 α+2β−1 (−1)n xcondition y(0) = ln . But if the terms ( ) and ( ) ( )n are treated independent to each 4 α+nβ−1 n 4 α+2β−1othern andβ arbitrarily parameter β is allowed to approach to zero only in the coefficient ( ) of (−1) x n 1 α+nβ−1( ) ( ) independently, the boundary condition y(1) = ln expressed in expanded series form n 4 5matches the prescribed value if it is imposed on y1 (x). Thus improvisation on y1 (x)in this way not onlyshoots to satisfy the other boundary condition but also exculpate to procures the exact solution. Therefore,allowing the process to do so and let the first iterate mend its way to produce exact solution y(x) = y1 (x)to the problem (5.3). Therefore the exact solution to (5.6) is given by 1 xβ 1 xβ (−1)n xβ 1 y(x) = y1 (x) = ln − + ( )2 )n +∑∞ ( n=3 )( )n =ln 4 4 2 4 n 4 4+xβ6. ConclusionIn this paper, we have applied the He’s variation iteration method successfully to a linear as well as to anonlinear class of boundary value problems. The convergence analysis of the proposed method withreference to considered class has also been presented in exhaustive manner. A proper selection of selectivefunction and careful imposition of boundary condition on iterative function may lead to an exact solutionor any other solution of high accuracy even to a non-linear problem in just only some maneuveredsimplifications.ReferencesS.Chandrasekher; Introduction to the study of stellar structure: Dover, New York 1967.A.S.Eddington; The Internal constitution of the stars: Cambridge University Press, London. N.Tosaka, S.Miyake; Numerical Approximation by an integral equation for the unsteady state heatconduction in the human head; J. College of industrial Technology: Nihon University 15(1982)69.D.S.Mc Elwain; A reexamination of oxygen diffusion in a spherical cell with Michaela-Menton KineticsJ.Theo.Bio. 71(1978)255.N. Andersion, A.M.Arthers; Complementary variational principles for diffusion problems with Michaelis-Menton Kinetics: Bull. Math Bio. 42(1980)131.Adam J.A. and Maggelakis S.A, Mathematical models of tumor growth IV Effects of a Necrotic core;Math.Bio. (1989) 121136.Pandey R.K., Verma A.K, Existence-Uniqueness results for a class of singular boundary value problemsarising in physiology, Nonlinear Analysis: Real World Application, 9(2008)40-52.W.F.Ford, J.A.Pennline; Singular nonlinear two-point boundary value problems; Existence and uniqueness:Nonlinear Analysis: 1(2009)1059-1072.M, M, Chawala; A fourth order finite difference method based on uniform mess for singular boundary valueproblems: Journal of Computational and applied mathematics 17 (1987)359-364.G.N.Reddien; Projection Method and singular two point boundary value problems: NumerischeMathematik 121.193-205. D01.10.1007/BFD 1436623.S.R.K.Iyenger, Pragya Jain; Spline finite difference methods for singular two- point boundary valueProblems; Numerishe Mathematik50 (1987) 363-376.Manoj Kumar; Higher order method for singular boundary value problems by using spline function:Journal of: Appl.Maths and Comput: 192(2007)17.A.S.V.Ravi Kanth, Y.N.Reddy; Cubic spline for a class of two-point boundary value problems. Appl. Mathsand Comput.170 (2005)733-740.Vedat Suat Erturk; Differential Transformation method for solving differential equation of Lane-Emdentype. Math. Comput. Appl. 12(2007)135-139.D.D.Ganji, A.Sadighi; Application of homotopy perturbation and variational iteration methods to nonlinearheat transfer and porous meadia equations: J.of Comput and Appl. Math: 207(1): (2007)24-34. 6
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