Numerical Solution of 12 th Order Boundary Value Problems by Using Homotopy Perturbation Method

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1 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 The Journal of athematics and Computer Science Available online at Journal of athematics and Computer Science Vol. No. () 4-7 Numerical Solution of th Order Boundary Value Problems by Using Homotopy Perturbation ethod ohamed I. A. Othman *, A.. S. ahdy and R.. Farouk 3,,3 Department of athematics, Faculty of Science, Zagazig University, P.O. Bo 4459, Zagazig, Egypt. * Department of athematics, Faculty of Science, Al-Dawadme, P.O. Bo 8, AL-Dawadmi 9, King Saud University, Kingdom of Saudi Arabia. 3 Department of Computer Science, Faculty of Science, Al-Dawadme, P.O. Bo 8, AL-Dawadmi 9, King Saud University, Kingdom of Saudi Arabia. m i othman@yahoo.com, amr mahdy85@hotmail.com and rmfarouk@yahoo.com Received: ay 9, Revised: Aug 9 Online Publication: Apr ABSTRACT In this paper, a homotopy-perturbation method (HP) [-6, 6-8] is used to solve both linear and nonlinear Nth boundary value problems with two point boundary conditions for ninth-order, tenth-order and twelfth-order. By applying (HP) on three eamples the numerical results are compared with the eact solution, to show effectiveness and accuracy of the method. Keywords: HP linear and non-linear problems boundary value problem Approimate solution. Introduction A new perturbation method called homotopy perturbation method (HP) was proposed by He in 997 and systematical description in which is, in fact, a coupling of the traditional perturbation method and homotopy in topology [-]. This new method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [3], nonlinear wave equations [4], asymptotology [5], boundary value problem [6], limit cycle and bifurcation of nonlinear problems [7] and many other subjects. Thus He s method is a universal one which can solve various kinds of nonlinear equations. For eample, it was applied to the quadratic Ricatti differential equation by Abbasbandy [8]; to the aisymmetric flow over a stretching sheet by Ariel et al. [9]; to the nonlinear systems of reaction-diffusion equations by Ganji and Sadighi []; to the Helmholtz equation and fifth-order KdV equation by Rafei and Ganji []; for the thin film flow of a fourth grade fluid down a vertical cylinder by Siddiqui et al. []; to the nonlinear Voltra-Fredholm integral equations by Ghasemi et al. [3]. In recent years, the application of the homotopy perturbation method (HP) [5, 7, 3] in nonlinear problems has been developed by scientists and engineers, because this method deforms the difficult problem under study into a simple problem which is easy to solve. ost perturbation methods assume a small parameter eists, but most nonlinear problem have no small at all. any new methods, such as the variational method [36-37], variational iterations method [4, 7] and (DT) [38-39]. The homotopy perturbation method proposed by He [, 6] is constantly being developed and applied to solve various nonlinear problems [3, 4, 7,, 7-3, 34]. Unlinke analytical perturbation methods, the homotopy-perturbation 4

2 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 th method does not depend on a small parameter which is difficult to find. We focus on dealing with the Ν boundary value problems by means of the homotopy perturbation method. Three numerical eamples will be presented to verify the homotopy perturbation method. Recently various powerful mathematical methods such as variational iteration method [4-3], ep-function method [4-5], f-epansion method [4], this paper, applies the homotopy perturbation method (HP) [-5].. Homotopy perturbation method th We consider the general Ν order boundary value problems of the type: (n) (n ) y () = f(,y,y (), K,y ()), < < () with the boundary conditions (k) y () =α k, k =,,,3 L,(n ) () (k) y () =β k, k =,,, 3 L,(n ) (3) (n ) where f(, y, y (), K, y ()) and y() are assumed real and as many as times differentiable as required for [,], α k and β k, k,,, L,(n ) are real finite constants Djidjedi, Twizell and Boutayeb [4], more over the constants k,k,,, L,(n ) describe the even order derivatives at the boundary Using the transformation n dy d y d y y = y, = y, = y,, y 3 L = n n d d d (4) We can rewrite the th Ν -order boundary value problems (), () and (3) as the system of ordinary differential equations: dy = y, d d y = y 3, d n d y = y n n, d dy = f(,y,y (),y (), L,y n ()). d (5) with the boundary conditions y (a) =α,y (a) =α, L,y n(a) =αn (6) or y (b) =β,y (b) =β, L,y n(b) =βn (7) which can be written as a system of integral equations: y =α + y (t)dt, y =α + y (t)dt, 3 y3 =α + y (t)dt, 4 yn =α n + f(t,y (t),y (t), L,y (t))dt. (8) n 5

3 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 To eplain (HP), we consider (8) as [, 7]: L(y,y, L,yn L(y,y, L, y n) = =, L n (y,y, L,yn with solution (f,f, L,f n) where (9) L(y,y, L,y n) = y α y (t)dt, We can define homotopy H(y,y, L,y n,p) by L n (y,y, L,y n ) = yn αn f(t,y (t),y (t), L,y (t))dt. n H(y,y, L,y n,) = F(y,y, L,y n,h(y,y, L,y n,) = L(y,y, L,y n), where T F(y,y, L,y n) = [F(y,y, L,y n), L,F n(y,y, L,y n)] T = [y α, L,y n αn,y n αn ]. () T H(y,y, L,y n,p) = [H (y,y, L,y n,p), L,H n (y,y, L,y n,p)]. () Typically we may choose a conve homotopy by: H(y,y, L,y n,p) = ( p)f(y,y, L,y n) + pl(y,y, L,y n) =. (3) The conve homotopy (3) contiuously trace an implicitly defined carve from a starting point H(y α, y αl, y n αn,) to a solution function H(f,f, L,f n,). The embedding parameter p monotonically increasing from zero to unite as trivial problem F(y,y, L,y n) = is continuously deformed to original problem L(y,y, L,y n) =. The (HP) uses the homotopy parameter p as an epanding parameter: y = y + py+ p y +L, yn = yn + pyn + p y n +L. (4) The approimate solution of Eq. (), therefor, can be readily obtained: f = lim y = y + y + y +L, p f n = lim y n p = y n + y n + y n +L. (5) The convergence of the series (4) for the application of (HP) to (8), we can write Eq. (3) as follows: H (y,y, L,y n,p) = ( p)f (y,y, L,y n) + pl (y,y, L,y n) =, H n (y,y, L,y n,p) = ( p)f n (y,y, L,y n ) + pl n (y,y, L,y n ) =. (6) substitution of (), () and (3) into (6) yields ( p)(y α ) + p(y α y (t)dt =, ( p)(y n α n ) + p(yn αn f(t,y (t),y (t), L,y (t))dt =. (7) n By equating the terms with identical powers of p, we have () 6

4 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 y =α, p : y =α, (8) y n =αn, y = y (t)dt, p : (9) yn, = y (t)dt, n d y n, = (f(t,y (t), L,y (t),y (t)))dt, n n dp y = y (t)dt, p : yn, = y (t)dt, () n d y n, = (f(t,y (t), L,y (t),y (t)))dt, n n dp y,k = y (t)dt,, k k p : yn,k = y (t)dt, () n, k k d y n,k = (f(t,y (t), L,y (t),y (t)))dt. k n n dp Combining all the terms of Eqs. (8)-() give the solution of the problem, by using the boundary conditions (6) and (7) we can obtained all parameters. 3. Applications In this section, in order to verify numerically whether the proposed methodology leads to higher accuracy, we evaluate the numerical solution of the problem. To show the efficiency of the present method for our problem in comparison with the eact solution we report absolute error which is defined by N E yn () = abs(y Eact () y Appro ()). () where N N y Appro () yk k = for N,,, L. (3) 4. Eamples Eample. Consider the following linear ninth-order problem (9) y () = y() 9e, a < < b. (4) with the following boundary conditions 7

5 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 (k) y () = ( k), k =,,,3,4. (5) (k) y () = ( ke), k =,,,3. (6) The eact solution is y() = ( )e. (7) Using the transformation (4) we can rewrite the ninth-order boundary value problem (4) as the system of integral equations: y = + y (t)dt, y = y (t)dt, 3 y3 = + y (t)dt, 4 y4 = + y (t)dt, 5 y5 = 3+ y (t)dt, 6 y6 = a + y (t)dt, (8) 7 y7 = b+ y (t)dt, 8 y8 = c+ y (t)dt, 9 t y9 = d + [ 9e + y (t)]dt. Using (7) into (8) we have y + py + p y + L= + p (y + py + p y + L )dt, y + py + p y + L= p (y + py + p y + L )dt, t y9 + py9 + p y9 + L= d+ p ( 9e + y + py + p y + L )dt. (9) Comparing the coefficients of like powers of p, we have: 8

6 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 y y =, y3 =, y4 =, p : y5 = 3, y6 = a, y7 = b, y8 = c, y9 = d. y =, y =, y3 =, y4 = 3, p : y5 = a, y6 = b, y7 = c, y 8 = d, y9 = 9e y =, y =, 3 y3 =, y4 = a, p : y5 = b, y6 = c, y7 = d, y8 = 9e , y9 =. (3) (3) (3) 9

7 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () y8 = d, y8 = 9e + d , y38 =, 8 y48 =, 8 43 p : 8 y58 =, 43 8 y68 =, y78 = a, 43 8 y88 = b, 43 8 y98 = c y9 = 9e , 8 y9 =, 9 y39 =, y49 =, p : 9 y59 =, 96 9 y69 = a, y79 = b, y89 = c, y99 = d Combining all the terms of Eqs. (3)-(34) gives (9) y () = a b + c + d. (35) Using the boundary conditions (6) then we have: a = , b = , c = , d = (9) y () = (36) (33) (34)

8 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 The numerical results obtained in Table.. In Table., we list the results obtained by homotopy perturbation method (HP) and compared with the eact solution. As we see from this Table, it is clear that the results obtained by the present method are very superior to that obtained by the eact solution highly accurate. Table.: Linear ninth-order BVP X HP(N=9) y = ( ) e..e+.e e e e e e e e e+..e+.e+ eact Eample. Consider the following linear tenth-order problem () y () = e y (), a < < b. (37) with the following boundary conditions (k) y () = (), k =,,,3,4. (38) (k) y () = (e), k =,,,3,4. (39) The eact solution is y() = e. (4) Using the transformation (4) we can rewrite the tenth-order boundary value problem (37) as the system of integral equations: y = + y (t)dt, y = a + y (t)dt, 3 y3 = + y (t)dt, 4 y4 = b+ y (t)dt, 5 y5 = + y (t)dt, 6 y6 = c+ y (t)dt, 7 y7 = + y (t)dt, 8 y8 = d+ y (t)dt, 9 y9 = + y (t)dt, t () y = f + [e y (t)]dt. (4)

9 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 Using (7) into (4) we have y + py + p y + L= + p (y + py + p y + L)dt, y + py + p y + L= a + p (y + py + p y + L)dt, t y + py + p y + L= f + p e (y + py + p y + L ) dt. (4) Comparing the coefficients of like powers of p, we have: y y9 = a, y3 y4 = b, y5 p : (43) y6 = c, y7 y8 = d, y9 y = f. y = a, y =, y3 = b, y4 =, y5 = c, p : y6 =, y7 = d, y8 =, y9 = f, y = e +. y =, y = b, y3 =, y4 = c, p : y5 =, y6 = d, y7 =, y8 = f, y9 = e +, y = ae ae +. (44) (45)

10 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () y3 = b, 6 3 y3 =, 6 3 y33 = c, 6 3 y43 =, p : y53 = d, 6 3 y63 =, 6 3 y73 = f, 6 y83 = e + +, y93 = ae + 4ae a 4a, y 3 = (+ a )( e e e + ). (46) Combining all the terms of (43)-(46) we get () y ()= a+ b + c + d + f (47) Using the boundary conditions (39) then we have: a =.933, b = , c =.8535, d = , f =.888. () y () = E( 5). (48) The numerical results obtained in Table.. In Table., we list the results obtained by homotopy perturbation method (HP) and compared with the eact solution. As we see from this Table, it is clear that the results obtained by the present method are very superior to that obtained by the eact solution highly accurate. As can be seen from Table., the error decreased when the integer N is decreased. Table.: Non-linear tenth-order BVP X HP (N=) y = e..e+.e e+.4758e e e e+.888e e e e e+ eact Eample 3. Consider the following linear twelve-order problem () (3) y () = e y () + y (), a < < b. (49) with the following boundary conditions 3

11 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 (k) y () k =,,,3,4,5. (5) (k) y () = ( ), k =,,,3, 4,5. (5) e The eact solution is y() = e. (5) Using the transformation (4) we can rewrite the twelve-order boundary value problem (49) as the system of integral equations: y = + y (t)dt, y = A+ y 3(t)dt, y3 = + y 4(t)dt, y4 = B+ y 5(t)dt, y5 = + y 6(t)dt, y6 = C+ y 7(t)dt, y7 = + y 8(t)dt, y8 = D+ y 9(t)dt, y9 = + y (t)dt, y = E+ y (t)dt, y = + y (t)dt, (53) t ( 3) y = F + [e y (t) + y (t)dt. Comparing the coefficients of like powers of p, we have: y y = A, y3 y4 = B, y5 y6 = C, p : y7 y8 = D, y9 y = E, y y = F, (54) 4

12 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 y = A, y =, y3 = B, y4 =, y5 = C, y6 =, p : y7 = D, y8 =, y9 = E, y =, y = F, y = e, y =, y = B, y3 =, y4 = C, y5 =, p : y6 = D, y7 =, y8 = E, y9 =, y = F, y = e, y = 4Ae 4Ae + 4A, Combining all the terms of (54)-(56) we obtain: () y () = A+ B + C + D + E + F (57) Using the boundary conditions (39) then we have: A = , B =.58885, C = , D =.5758, E = , F = () y () = E( 5) E( 7) (58) The numerical results obtained in Table.3. In Table.3, we list the results obtained by homotopy perturbation method (HP) and compared with the eact solution. As we see from this Table, it is clear that the results obtained by (55) (56) 5

13 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 the present method are very superior to that obtained by the eact solution highly accurate. As can be seen from Table.3, the error decreased when the integer N increased. Table.3: Non-Linear twelfth-order BVP X HP (N=) y eact = e..e-.e E E E E E E E E E E- 4. Conclusion Homotopy perturbation method is applied to the numerical solution for solving both linear and nonlinear boundary value problems. Comparison of the results obtained by the present method with that obtained by eact solution reveals that the present method is very effective and convenient. The numerical results in the Tables [.-.3], show that the present method provides highly accurate numerical results. This method cannot be applied easily if the order of equation is higher than th order. th N 5. References [] A. Golbabai,. Javidi, Application of homotopy perturbation method for solving eighth-order boundary value problems, Appl. ath. Comput., 9() (7) [] J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non-Linear ech., 35() () [3] J. H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. ath. Comput., 5() (4) [4] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 6(3) (5) [5] J. H. He, Asymptotology by homotopy perturbation method, Appl. ath. Comput., 56 (3) (4) [6] J. H. He, Homotopy perturbation method for solving boundary problems, Phys. Lett. A, 35() (6) [7] J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals, 6(3) (5) [8] S. Abbasbandy, Iterated Hes homotopy perturbation method for quadratic Riccati differential equation, Appl. ath. Comput., 75() (6) [9] P. D. Ariel, T. Hayat, S. Asghar, Homotopy perturbation method and aisymmetric flow over a stretching sheet, Int. J. Nonlinear Sci. Numer. Simul., 7(4) (6) [] D. D. Ganji, A. Sadighi, Application of He s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int. J. Nonlinear Sci. Numer. Simul., 7(4) (6) []. Rafei, D. D. Ganji, Eplicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul., 7(3) (6) [] A.. Siddiqui, R. ahmood, Q. K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A. 35(45) (6) [3] J. Rashidinia, Z. ahmoodi,. Ghasemi., Numerical solution of the nonlinear Voltra-Fredholm integral equations by using homotopy perturbation method, Appl. ath. Comput., 88() (7) [4] J. H. He, Approimate solution of nonlinear differential equations with convolution product nonlinearities, Comput. eth. Appl. ech. Eng., 67() (998) [5] J. H. He, Approimate analytical solution for seepage flow with fractional derivatives in porous media, Comput. eth. Appl. ech. Eng., 67() (998) [6] J. H. He, Variational iteration method kind of non-linear analytical technique: some eamples, Int. J. Non- Linear ech., 34(4) (999) [7] J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. ath. Comput., 4(-3) () 5-3. [8] S. omani, S. Abuasad, Application of He s variational iteration method to Helmholtz equation, Chaos Solitons Fractals, 7(5) (6) -93. [9] A. A. Soliman, A numerical simulation and eplicit solutions of KdV Burgers and Las seventh- order KdV equations, Chaos Solitons Fractals, 9() (6)

14 ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 [] E.. Abulwafa,. A. Abdou, A. A. ahmoud, The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals, 9() (6) [] Z.. Odibat, S. omani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7() (6) []. Javidi, A. Golbabai, Eact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Solitons Fractals, 36() (8) [3] I. H. Abdel-Halim Hassan, ohamed I. A. Othman, A.. S. ahdy, Variational iteration method for solving twelve order boundary value problems, Int. J. ath. Analysis, 3(5) (9) [4] J. H. He, Ep-function method for nonlinear wave equations, Chaos Solitons Fractals, 3(3) (6) [5] J. H. He,. A. Abdou, New periodic solutions for nonlinear evolution equations using Epfunction method, Chaos Solitons Fractals, 34(5) (7)4-49. [6] J. H. He, New interpretation of homotopy perturbation method, Internat. J. odern Phys. B, (8) (6) [7] J. H. He, Homotopy perturbation technique, Comput. ath. Appl. ech. Eng, 78(3-4) (999) [8] J. H. He, Homotopy perturbation method: A new nonlinear technique, Appl. ath. Comput, 35 (3) [9] J. H. He, Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347 (5) 8-3. [3] J. H. He, Homotopy-perturbation method for bifurcation of nonlinear problems, Internat. J. Nonlinear Sci. Numer. Simul, 6 (5) 7-8. [3] L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals, 3(5) (6)-3. [3] S. Abbasbandy, Application of He s homotopy perturbation method for Laplace transform, Chaos Solitons Fractals, 3(5) (6) 6-. [33] A.. Siddiqui, R. ahmood, Q.K. Ghori, Thin film flow of athird grade fluide on a moving belt by He s homotopy perturbation method, Internat. J. Nonlinear Sci. Numer. Simul, 7() (6) 7-4. [34] A.. Siddiqui,. Ahmed, Q.K. Ghori, Couette and Poiseuille flows for non-newtonian fluids, Internat. J. Nonlinear Sci. Numer. Simul, 7() (6) 5-6. [35] S. J. Liao, An approimate solution technique not dependind on small parameters: A special eample, Int. J. Non- Linear ech., 3(3) (995) [36] J. H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals, 9(4) (5) [37] H.. Liu, Variational approach to nonlinear electrochemical system, Chaos Solitons Fractals, 3() (5) [38] I. H. Abdel-Halim Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons and Fractals, 36 (8) [39] I. H. Abdel-Halim Hassan, Application to differential transformation method for solving systems of differential equations, Appl. ath. odell., 3() (8) [4] K. Djidjedi, E. H. Twizell, A. Boutayeb, Numerical methods for special non linear boundary value problems of order m., J. Comput. Appl. ath., 47 (993)