Numerical solution of fourth order parabolic partial dierential equation using parametric septic splines

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1 Hacettepe Journal of Mathematics and Statistics Volume 5 20, Numerical solution of fourth order parabolic partial dierential equation using parametric septic splines Arshad Khan and Talat Sultana Abstract In this paper, we report three level implicit method of high accuracy schemes for the numerical solution of fourth order nonhomogeneous parabolic partial dierential equation, that governs the behavior of a vibrating beam. Parametric septic spline is used in space and nite dierence discretization in time. The linear stability of the presented method is investigated. The computed results for three examples are compared wherever possible with those already available in literature which shows the superiority of the proposed method. Keywords: Parametric septic splines; Fourth order parabolic equation; Stability analysis; Vibrating beam; Finite dierence scheme. Received : Accepted : Doi : 0.572/HJMS Introduction In this paper, we consider the problem of undamped transverse vibration of a exible straight beam in such a way that its support do not contribute to the strain energy of the system and is represented by the fourth order parabolic partial dierential equation of the form 2 u t u = fx, t, a x b, t > 0,. 2 x subject to the initial conditions ux, 0 = g 0x, a x b, u tx, 0 = g x, a x b Department of Mathematics, Jamia Millia Islamia, New Delhi-25, INDIA. akhan23in@rediffmail.com Department of Mathematics, Lakshmibai College, Univ. of Delhi, New Delhi-52, INDIA. talat7m@gmail.com.2

2 08 and the boundary conditions ua, t = f 0t, ub, t = f t, t 0, u xxa, t = q 0t, u xxb, t = q t, t 0,.3 where u is the transverse displacement of the beam, g 0x, g x, f 0t, f t, q 0t, q t are continuous functions, t and x are time and distance variables respectively and fx, t is dynamic driving force per unit mass [0,2,22,35. Numerical methods for the solution of equation. have been carried out by many authors. Jain et al. [2, Danaee and Evans [, Evans [8, Collatz [20, Andrade and Mckee [7 and Evans and Yousif [9 used nite dierence methods for the numerical solution of transverse vibrations. Fairweather and Gourlay [3 derived explicit and implicit nite dierence methods based on the semi explicit method. Parametric quintic spline methods are given by Rashidinia and Aziz [ using nodal points. Collatz [20, Crandall [33, Jain [23, Conte [32, Jain et al. [2 and Todd [8 have proposed both explicit and implicit methods successfully. Five level, unconditionally stable, explicit method with truncation error of Ok 2 h 2 k h 2 has been given by Albrecht [5. All the above authors considered the homogeneous case of equation. with a constant coecients. The analytical solution of homogeneous case of equation. has been obtained by using Adomain decomposition method by Wazwaz [3,. The nonhomogeneous problem with constant coecients has been studied by Aziz et al. [3 based on parametric quintic spline and by Khan et al. [2 based on sextic spline by using nodal points. Khaliq and Twizell [ and Twizell and Khaliq [ developed a family of numerical methods, which are second order accurate in space and time, based on exact recurrence relation for a homogeneous case of equation. with a variable coecient. Rashidinia and Mohammadi [7 developed three level implicit methods of Ok 2 h and Ok h for the numerical solution of equation. with variable coecients by using sextic spline. Wazwaz [5 has developed analytical solution of variable coecient fourth order parabolic partial dierential equation in two and three space dimensions. Khan et al. [25 have introduced a new algorithm, namely Laplace Decomposition Algorithm for fourth order parabolic partial dierential equations with variable coecients. In [2, the homotopy analysis method HAM is applied to solve such problems. Khan et al. [27 have studied numerical solution of time fractional fourth order partial dierential equations with variable coecients. They have implemented reliable series solution techniques namely, Adomian Decomposition Method ADM and He's Variational Iteration Method HVIM. A family of B-spline methods have been considered by Caglar [. In [28, Mittal and Jain discussed two methods. In Method-I, they decomposed equation. in a system of second order equations and have solved them by using cubic B-spline and in Method-II, they have solved equation. directly by using quintic B-spline method. Talwar et al. [9 and Mohanty et al. [29-3 have used high accuracy spline scheme for solving one dimensional partial dierential equations. In this paper, parametric septic spline relations have been derived using nodal points. We have used parametric septic spline functions to develop a new numerical method for obtaining smooth approximations to the solution of nonhomogeneous parabolic partial dierential equations dealing with vibrations of beams. In section 2, parametric septic spline and spline relations are developed. In section 3, we have presented the formulation of our method. Development of boundary equations are given in section. In section 5, truncation error and class of methods are given. Stability analysis is discussed in section. Finally in section 7, three examples are given to demonstrate the practical usefulness and superiority of our method.

3 09 2. Parametric septic spline Let a set of grid points in the interval [a, b such that x j = a jh, j = 0N, h = b a N. 2. A function S x, τ of class C [a, b which interpolates ux at the mesh point x j depends on a parameter τ, and as τ 0 it reduces to septic spline S x in [a, b is termed as parametric septic spline function. Since the parameter τ can occur in S x in many ways such a spline is not unique. If S x, τ = S x is a piecewise function satisfying the following dierential equation in the interval [x j, x j S x τ 2 S x = Q j τ 2 M x j xj Q j τ 2 M j xj x h h = A jz A j z, 2.2 where z = x xj, z = z, A i = Q i τ 2 M i, h S x i, τ = M i, S xi, τ = Qi, i = j, j; τ > 0, then it is termed as parametric septic spline II. Solving equation 2.2, we get S x = A A 2x A 3 cosh τx A sinh τx A 5 cos τx A sin τx } {Q τ 2 j τ 2 x xj 3 M j Q j τ 2 xj x3 M j h h 2.3 To develop the consistency relations between the value of spline and its derivatives at knots, let S x j = u j, S x j = u j, S x j = M j, S x j = M j, S xj = Fj, S xj = Fj. 2.

4 070 To dene spline in terms of u j's, M j's and F j's, the coecients introduced in Eq.2.3 are calculated as A = u j h2 τ 2 Qj τ 2 M j Fj τ 2 xj h A 2 = h uj uj h τ 2 A 3 = [u j u j h2 τ 2 Qj τ 2 M j h2 [ τ 2 sinh τh 2 sinh τx j [ Q j τ 2 M j Q j τ 2 M j F j Qj τ τ sinh τx j Q j τ sinh τx jq j, Fj Fj, 2 τ 2 Qj τ 2 M j τ τ 2 Fj Fj, h 2 sinh τx j F j Qj τ A = [ τ 2 sinh 2 τh cosh τx j F j Qj 2 τ cosh τx j F j Qj τ A 5 = A = τ cosh τx j Q j τ cosh τx jq j, 2τ 2 sinh τh 2τ 2 sinh τh [sin τx j F j Qj τ [ cos τx j F j Qj τ sin τx j F j Qj τ, cos τx j F j Qj τ Substituting these values in 2.3, we get S x = zu j zu j [pzm h2 j p zm j [rzf h j r zf j 2 [qzq h j q zq j, where p z = z 3 z, q z = z ω z3 3 sinh ωz 3 sin ωz ω ω sinh ω ω sin ω, r z = 2z sinh ωz ω ω sinh ω sin ωz ω sin ω and ω = τh. 2.7 Applying the rst, third and fth derivative continuities at the knots, i.e. S µ x j = S µ x j, µ =, 3 and 5, the following consistency relations are derived: M j M j M j = h 2 uj 2uj uj 3h2 α 2F j 2β 2F j α 2F j h α Q j 2β Q j α Q j, j = N. 2.8 M j 2M j M j = h2 [ ω α F j 22 ω β F j ω α F j h α2qj 2β2Qj α2qj, j = N

5 07 where h 2 [ ω α Q j 22 ω β Q j ω α Q j = 3[ω α 2 2F j 2ω β 2 2F j ω α 2 2F j, j = N, α = ω 3 ω 5 sinh ω 3 ω 5 sin ω, β = 2 ω 3 ω coth ω 3 cot ω, 5 ω5 α 2 = 2 ω ω 3 sinh ω ω 3 sin ω, 2.0 β 2 = 2 ω ω coth ω cot ω ω3 As τ 0 that is ω 0 then α, β, α 2, β 2 3,, 7, Using equations , we obtain the following scheme e u j 3 e 2u j 2 e 3u j e u j e 3u j e 2u j2 e u j3 = h pfj 3p2Fj 2p3Fj pfjp3fjp2fj2pfj3, j = 3N 3, 2.2 where the coecients e, e 2, e 3, e and p, p 2, p 3, p of the developed scheme are given by e = 3ω α 3ω 8 α 2 ω 2 α 3, e 2 = ω α 2ω β 8ω 8 α 2 ω 8 α β 2ω 2 α 2 β, e 3 = 7 ω α 3 8 ω α 2 2 ω β, e = 2 ω α 2 2 ω β 8 ω α 3, p = c ω α 2, p 2 = 2c ω α 2 ω β c 2 ω α 2 3d ω α 2 ω α 2, p 3 = c c 3 ω α 2 d ω α 2 ω β 2 2c 2 ω α 2 ω β 3d 2 ω α 2 ω α 2, p = 2c 2 ω α 2 d ω α 2 ω α 2 d 2 ω β 2 ω β 2 Also 2c 3 ω α 2 ω β d 2 ω α 2 ω β 2. c = ω8 α ω α ω α α α 2, 2.3 c 2 = 2 3 ω8 α 2 3 ω8 α β 8ω α α 2 3ω α 2β 2 ω α 2 2 2ω α 3 ω β 2α β 2 3, c 3 = 3 ω8 α 2 3 ω8 α β 3ω α β 2 2ω α 2β 2 3ω α ω α 3 ω β 3α 2α 2 2β 2 3, d = ω α 2β ω α β 2 ω α 2 0α 2α 2 2β β 2, d 2 = ω α 2β ω α β 2 2ω α β α 8α 2 β β 2. 2.

6 072 As τ 0 that is ω 0, we have ie, e 2, e 3, e, 0, 9,, iic, c 2, c 3, d, d 2 0, 7, 29 70, 9 0,, 35 iiip, p 2, p 3, p 0, 7, 9 0, [Remarks: For these values our scheme reduces to the polynomial septic spline for fourth order boundary value problem which is given as equation 7 in G. Akram and S. S. Siddiqi [2. Here, we have taken e, e 2, e 3, e =, 0, 9,, therefore scheme 2.2 becomes p F j 3 F j3 p 2F j 2 F j2 p 3F j F j p F j = h [ u j 3 u j3 9u j u j u j, j = 3N We can also write 2.5 as Λ xf j = h δ x δ xu j, 2. where δ is the central dierence operator and operator Λ x for any function W is dened by Λ xw j = p W j 3 W j3 p 2W j 2 W j2 p 3W j W j p W j Derivation of the method Let the region R = [a, b [0, be discretized by a set of points R h,k which are the vertices of a grid points x j, t m, where x j = jh, j = 0N, Nh = b a and t m = mk, m = 0,, 2, 3,... The quantities h and k are mesh sizes in the space and time directions respectively. We have developed an approximation for. in which the time derivative is replaced by a nite dierence approximation and space derivative is replaced by the parametric septic spline function approximation. We need the following nite dierence approximation for the time partial derivative of u : u m tt j = k 2 δ 2 t σδ 2 t u m j, 3. where σ is a parameter such that the nite dierence approximation to the time derivative is Ok 2 for arbitrary σ and Ok for σ = /2. u m j is the approximate solution of. at x j, t m and δ t is the central dierence operator with respect to t so that δ 2 t u m j = u m j 2u m j u m j. At the grid point j, m the dierential equation may be discretized by u m tt j u m xxxx j = f m j, 3.2 where u m xxxx j is the fourth order spline derivative at x j, t m denoted by Fj m = S xj, tm with respect to the space variable fj m = fx j, t m. Using 3. and replacing fourth order spline derivative by Fj m, we have k 2 δ 2 t σδ 2 t u m j F m j = f m j. 3.3 Operating Λ x on both sides of 3.3 and using 2., we obtain δ 2 t [p u m j 3 u m j3p 2u m j 2 u m j2p 3u m j u m jp u m j r 2 σδ 2 t δ x δ xu m j = k 2 σδ 2 t [p f m j 3 f m j3p 2f m j 2 f m j2p 3f m j f m jp f m j, j = 3N 3, 3.

7 073 where r = k is the mesh ratio and p h, p 2, p 3, p 2 are parameters. After simplifying the above equation, we obtain δ 2 t [2p 2p 2 2p 3 p 9p p 2 p 3δ 2 x p p 2 3σr 2 δ x p σr 2 δ xu m j r 2 δ xδ xu m j = k 2 σδ 2 t [p f m j 3f m j3p 2f m j 2f m j2p 3f m j f m jp f m j, j = 3N This scheme 3.5 is nite dierence in time and spline scheme in space variable, which on simplication can be written as [P u m j 3 u m j3 P 2u m j 2 u m j2 P 3u m j u m j P u m j [S u m j 3 u m j3 S 2u m j 2 u m j2 S 3u m j u m j S u m j [P u m j 3 u m j3 P 2u m j 2 u m j2 P 3u m j u m j P u m j = K [p f m j 3 f m j3 p 2f m j 2 f m j2 p 3f m j f m j p f m j K 2[p f m j 3 f m j3 p 2f m j 2 f m j2 p 3f m j f m j p f m j K [p f m j 3 f m j3 p 2f m j 2 f m j2 p 3f m j f m j p f m j, j = 3N The nal scheme 3. may be written in the schematic form as P P 2 P 3 P P 3 P 2 P K p K p 2 K p 3 K p K p 3 K p 2 K p S S 2 S 3 S S 3 S 2 S u m j = K 2p K 2p 2 K 2p 3 K 2p K 2p 3 K 2p 2 K 2p P P 2 P 3 P P 3 P 2 P where K p K p 2 K p 3 K p K p 3 K p 2 K p P = p σr 2, P 2 = p 2, P 3 = p 3 5σr 2, P = p 9σr 2, S = 2P σr 2, S 2 = 2P 2, S 3 = 2P 3 5σr 2, S = 2P 9σr 2, K = σk 2, K 2 = k 2 2σ. fj m,. Development of boundary equations The relation 3. gives N 5 linear algebraic equations in N unknowns u j, j = 3N 3. We need four more equations, two at each end of the range of integration, for the direct computation of u j, j = N. i um um um um 32 7 um um = 9 um h2 u m 0, j =, ii um um um um 90 7 um um 7 um 7 = h2 u m 0, j = 2, iii 7 um N um N 90 7 um N um N um N um N um N = h2 u m N, j = N 2, iv 58 3 um N 32 7 um N um N um N um N um N = 9 um N 80 7 h2 u m N, j = N.

8 07 For high accuracy formula of Ok h 0, we use the following equations for approximating the boundary equations: i um 5 5 um um 3 u m um um 99 7 um um 8 = um h2 u m 0, j =, ii um um um um um um um um um 9 = h2 u m 0, j = 2, iii um N um N um N um N um N um N um N um N um N = h2 u m N, j = N 2, iv um N um N um N um N 5 8 u m N um N um N um N = um N h2 u m N, j = N. 5. Truncation error and class of methods Expanding 3.5 in Taylor series in terms of ux j, t m and its derivatives, we obtain the following relations δ xux j, t m = δ xux j, t m = δ 2 t ux j, t m = [ h D x ! [ h Dx 20 h 8 Dx h 0 Dx ! 2! ux j, t m h D x... h 2 D 2 x x...! h Dx 505 h8 Dx 8 0 0! h0 Dx ! h2 D 2 [ r 2 h Dx 2 r h 8 Dx 8 30 r h 2 Dx r8 h Dx... ux j, t m ux j, t m, 5.

9 075 where Dt 2 Dxux j, t m = fx j, t m. Using 3.5 and 5., we obtain the truncation error [ Tj m = 2p 2p 2 2p 3 p 9p p 2 p 3δx 2 p p 2 3σr 2 δx p σr 2 δx δt 2 u m j r 2 δx δxu m j [ k 2 σδt 2 p fj 3 m fj3 m p 2fj 2 m fj2 m p 3fj m fj m p fj m = [o k 2 Dt 2 2k Dt 2k Dt 2k8 Dt 8...!! o 2h 2 Dx 2 k 2 Dt 2 2k Dt! 2k D t! 2o 2 2o 3 h Dx k 2 Dt 2 2k Dt!! 2o 2 20o 3 720o h D x! 2o 2 50o o h8 D 8 x 2o 2 0o o h0 Dx 0 0! r 2! h Dx 0! [... u m j 2k8 D 8 t 2k D t! k 2 Dt 2 2k Dt! h D x 032 k 2 Dt 2 2k Dt!... 2k8 D 8 t 2k D t! k 2 D 2 t 2k D t! 2k D t!... 2k8 D 8 t 2k D t! h 8 Dx 8 93 h 0 D 0 0! 2k8 D 8 t... 2k8 D 8 t... x ! o k 2 o σk D 2 t 2o! σk D t 2o! σk8 D t 2o σk0 D 0 t... o 2h 2 Dx 2 28p p 2 p 3 h Dx! 25p 25p 2 p 3 h8 Dx 8... k 2 σk Dt 2 2σk Dt 2σk8 Dt!! 2729p p 2 p 3 h D x! 2σk0 D 8 t which may be written as [ Tj m = 3 o r 2 h Dx 2 o 2r 2 h Dx p p2 p3 r 2 h 0 Dx 0... Dt 2 Dxu m j 5 8p p2 p3 2 5p 25p2 p3 r 2 h 2 Dx σ o k Dt 2 σ o 2h 2 k DxD 2 t 30 σ 2 h 2 o2 2o3 oσk2 h 8p p2 p3 k DxD t 2 2 h 30 o2 0o3 30o o2σk2 σh 729p p2 p u m j h 2 D 2 x r 2 h 8 Dx 8 o 2h 2 k D 2 xd t h 2 k DxD t 2

10 07 h 200 σ h o2 2o3 2 oσk2 σh 2 o k 8 Dt σ 2 2σh h..., 8p p2 p3 k DxD t o k Dt 2 2σ o 2h 2 k 8 DxD 2 t 8! σk2 2o2 50o3 0080o 8p p2 p3 2 5p 25p 2 p 3 h k 2 DxD 8 t 2 o2 2o oσk2 σh 8p p2 p3 k DxD t... u m j 5.2 where o = 2p 2p 2 2p 3 p, o 2 = 9p p 2 p 3, o 3 = p p 2 3σr 2, o = p σr 2, D x = x, Dt = t. 5.3 For various values of parameters p, p 2, p 3, p and σ, we obtain the following class of methods: Case : If 3 o = 0, we obtain various schemes of Ok 2 h 2 for arbitrary values of σ. Case 2: If 3 o = 0 and 2 o 2 = 0, we obtain various schemes of Ok 2 h for arbitrary values of σ. Case 3: If 3 o = 0, 2 o 2 = 0, and 3 8p p2 p3 = 0, we obtain various 5 2 schemes of Ok h 8 for σ and 2 Ok h 8 for σ =. 2 Case : For p, p 2, p 3, p, σ =, 2, 8,,, we obtain a scheme of Ok h Stability analysis To investigate the stability analysis of the scheme 3., we use the Von Neumann method. We have assumed that the solution of 3. at the grid point x j, t m is of the form u m j = ξ m e jiθ,. where i =, θ is real and ξ in general is complex. Substituting. in homogeneous part of 3., we obtain a characteristic equation where Xξ 2 Y ξ Z = 0,.2 X = Z = P cos 3θ P 2 cos 2θ P 3 cos θ 2P, Y = S cos 3θ S 2 cos 2θ S 3 cos θ 2S. Under the transformation ξ = η, equation.2 becomes η X Y Zη 2 2X Zη X Y Z = 0..3 The necessary and sucient condition for ξ is that X Y Z > 0, X Z > 0 and X Y Z > 0. The conditions X Z > 0 and X Y Z > 0 are always satised for all real values of θ. From the condition X Y Z > 0, we get that the scheme 3. is unconditionally stable if σ and conditionally stable if σ < for all real values of p, p2, p3, p and θ. We summarized the above results in the following theorem:

11 077 Theorem: The scheme 3. for solving. is unconditionally stable if σ and conditionally stable if σ <. By using the Lax theorem, we can conclude that the present method is converge as long as stability criterion is satised. 7. Numerical results and discussions We have applied the presented method on the fourth order parabolic partial dierential equation and have considered one homogeneous and two nonhomogeneous examples. The proposed method 3. is implicit three level method based on parametric septic spline function. Example : Consider a nonhomogeneous fourth order parabolic partial dierential equation [2,9,7,3 2 u t u 2 x = π sin πx cos t, 0 x, t 0, subject to the initial conditions and the boundary conditions ux, 0 = sin πx, u tx, 0 = 0, 0 x u0, t = u, t = u xx0, t = u xx, t = 0, t 0. The analytical solution for this example is ux, t = sin πx cos t. We have solved the above example with h = 0.05 and k = giving r = 2 and by choosing σ =, 2 with Ok h 8, Ok h 8 and Ok h 0 for arbitrary choices of parameters p, p 2, p 3 and p. All computations have been done over ten time steps. The absolute errors at particular points x = 0., 0.2, 0.3, 0., 0.5 and comparison with other existing methods [2,8,5,2 are tabulated in table. We repeat the computations for time steps with r = 0.5.

12 078 Table. Absolute errors for example Methods r Time steps x = 0. x = 0.2 x = 0.3 x = 0. x = 0.5 p, p 2, p 3, p, σ Ok h Ok h , 9, 0,, Ok h , 9, 0,, [7 Ok 2 h, σ =, Ok h, σ =, [ [ [ Example 2: Consider a homogeneous fourth order parabolic partial dierential equation [8,33,3 2 u t u = 0, 0 x, t 0, 2 x subject to the initial conditions and the boundary conditions ux, 0 = x 2 2x2 x 3, u tx, 0 = 0, 0 x u0, t = u, t = u xx0, t = u xx, t = 0, t 0. The analytical solution for this example is ux, t = d s sin2s πx cos2s 2 π 2 t, where s=0 d s = 8 [2s 5 π 5.

13 079 We have solved this example with h = 0. for σ =,. The absolute errors at particular 2 points x = 0., 0.2, 0.3, 0., 0.5 with r = 2 and 50 time steps of Ok h 8, Ok h 8 and Ok h 0 for arbitrary choices of parameters p, p 2, p 3 and p and comparison with other existing methods are tabulated in table 2. We repeat the computations for 00 time steps with r =, 7 and r =. We have also included results given by 0 8 unconditionally stable method. Table 2. Absolute errors for example 2 Methods r 2 Time steps x = 0. x = 0.2 x = 0.3 x = 0. x = 0.5 p, p 2, p 3, p, σ Ok h Ok h , 9, 0,, Ok h , 9, 0,, [8, σ = [3, σ = [33, σ = [3, σ = Example 3: Consider a nonhomogeneous fourth order parabolic partial dierential

14 080 equation [32 2 u t u 2 x = [2 x2 x 2 cos t, 0 x, t > 0, subject to the initial conditions and the boundary conditions ux, 0 = x 2 x 2, u tx, 0 = 0, 0 x u0, t = u, t = 0, u xx0, t = u xx, t = 2 cos t, t 0. The analytical solution for this example is ux, t = x 2 x 2 cos t. We have solved this example with h = 0.05 for σ =,. The absolute errors at particular points x = 0., 0.2, 0.3, 0., 0.5 with r = 2 and 0 time steps for Ok h 8, Ok h 8 2 and Ok h 0 using arbitrary choices of parameters p, p 2, p 3 and p are tabulated in table 3. We repeat the computations for time steps with r = 0.5. Table 3. Absolute errors for example 3 Methods r Time steps x = 0. x = 0.2 x = 0.3 x = 0. x = 0.5 p, p 2, p 3, p, σ Ok h Ok h , 9, 0,, Ok h , 9, 0,, Conclusion The parametric septic spline function have been developed to obtain three level implicit methods for solving fourth order parabolic partial dierential equations. The developed methods are tested on three examples. The performance of these methods have been examined by comparing solution of homogeneous and nonhomogeneous fourth order parabolic partial dierential equations with available results. In examples, 2 and 3, we have computed absolute errors at the points x = 0., 0.2, 0.3, 0., 0.5 for the sake of comparison with our references and results are tabulated in tables -3. Tables show that our

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