Direct Explicit Integrators of RK Type for Solving Special Fourth-Order Ordinary Differential Equations With An Application

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume, Number 6 (06), pp Research India Publications Direct Explicit Integrators of RK Type for Solving Special Fourth-Order Ordinary Differential Equations With An Application Mohammed S. Mechee Department of Mathematics, Faculty of Computer Science & Mathematics, Kufa university, Najaf, Iraq. Murtaza A. Kadhim Department of Mathematics, Faculty of Computer Science & Mathematics, Kufa university, Najaf, Iraq. Abstract The main contribution of this paper is the development of direct explicit integrators of Runge-Kutta type for solving special fourth-order ordinary differential equations (ODEs) which is denoted as RKM. For this purpose, we generalized the integrators of Runge-Kutta type for solving special first, second and third-order ODEs (RK, RKN & RKD). Using Taylor expansion, we have derived the order conditions for the proposed RKM integrators, up to the seventh order. Based on these order conditions, three RKM methods of orders four, five and six with different stages are derived. Numerical implementation shows that the new methods agree well with existing RK methods but require less function evaluations. This is so due to the fact that RKM methods are direct; hence, they save considerable amount of computational time. An application for RKM method has been introduced for solving the mathematical model of beam s deflection. AMS subject classification: 05C78. Keywords: Runge-Kutta method; RK, RKN, RKD Integrators; fourth-order; Ordinary differential equations, Application of beam model.

2 4688 Mohammed S. Mechee and Murtaza A. Kadhim. Introduction The most important mathematical model for physical phenomena is the differential equation. Motion of objects, Fluid and heat flow, bending and cracking of materials, vibrations, chemical reactions and nuclear reactions are all modeled by systems of differential equations. Moreover, Numerous mathematical models in science and engineering are expressed in terms of unknown quantities and their derivatives. Many applications of differential equations (DEs), particularly ODEs of different orders, can be found in the mathematical modeling of real life problems. Fourth-order ODEs & PDEs occur in the physics and engineering ([6], [6], [9], [38], [9], [39], [4] and [33]). In addition to this type of Fourth-order DEs often arise in many fields of applied science such as mechanics, quantum chemistry, electronic and control engineering. However, Fourth order ODEs arise in several fields such as fluid dynamics ([]), beam theory ([4] and [5]), electric circuits ([]), ship dynamics ([40], [35] and [6]), neural networks ([0]), and the mathematical model of beam s deflection ([37] & [4]). Finding the solutions to these differential equations had been challenged the ingenuity of mathematicians since the time of Newton, thereby resulting in several powerful analytical techniques now available to modern scientists. Therefore, many theoretical and numerical studies dealing with the solution of such differential equations of different order have appeared in the literature. There are many analytical and numerical methods for solving some types of the differential equations. Unfortunately analytical tools frequently are inadequate for the solution of such systems. The number of cases where an exact solution can be found by analytical means is very limited, the only general class of systems for which exact solutions can always be found consists of linear systems with constant coefficients. However, in applications it is not necessary to find the solution to more than a finite number of decimal places. For this reason numerical methods were developed for the solution of ordinary differential equations since the age of Newton, Taylor and Euler. There are also many examples of particular linear variable coefficient or nonlinear systems for which exact solutions are known, but in general for such systems we must resort to either an approximate or a numerical method. These methods of solutions are not able to solve many types of differential equations or they can solved some types of differential equations indirectly. This reason make us to study and derive more direct numerical methods for this propose. In order to apply indirect numerical method to solve a differential equation of higher than order one, the equation should be transformed into a system of first order differential equations. Many researchers developed the family of Runge Kutta methods for solving first, second and third order ordinary differential equations, For example [3] has developed a singly diagonally implicit Runge-Kutta-Nyström (RKN) method for second-order ordinary differential equations with periodical solutions. Many applications has been solved base Runge Kutta methods. [] solved discrete-time model representation for biochemical pathway systems based on Runge Kutta method. One advantage of this approach is to improves the discretization accuracy by utilizing multi-step information for approximation of one-step-ahead model prediction. Moreover, it gives a general exact discrete-time representation for both linear and nonlinear biochemical ODE models.

3 Solving Special Fourth-Order Ordinary Differential Equations 4689 Another application in the field of visual Landmark selection for mobile robot navigation evaluated based on Runge Kutta methods for finding optimal solutions requires to solve some hard problems. [0] consider the minimum overlapping region decomposition problem that was proposed for landmarks selection. He was describe an approach to solve the problem optimally. This approach is based on an explicit reduction from the problem to the satisfy ability problem. The mathematical problems often arise in engineering and applied sciences such as celestial mechanics, quantum mechanics, elastic dynamics, theoretical physics, chemistry and electronics and can be solved by using Runge Kutta methods and multistep methods. It is important to increase the order of the methods to achieve higher accuracy. [8] derived some efficient methods for solving second order ordinary differential equations which have oscillating solutions, it is also essential to consider the phase-lag and the dissipation error that result from comparing. The objectives of this work are twofold. Firstly, we seek to establish direct explicit integrators of Runge-Kutta type for solving special fourth-order ordinary differential equations (ODEs). For this purpose, we have generalized the integrators of Runge- Kutta type for solving special first, second and third-order ODEs (RK, RKN & RKD). Secondly, we have compared the new direct integrators with the existing indirect RK methods.. Preliminary.. Special Fourth-Order Ordinary Differential Equations There are special fourth-order ODEs with no explicit dependence on the first, second and third derivatives y (x), y (x), and y (x). Such ODEs are frequently found in many physical and engineering problems. They can be written in the following form: with initial conditions, where and y (4) (x) = f(x,y(x)), x x 0, (.) y(x 0 ) = α 0,y (x 0 ) = α,y (x 0 ) = α and y (x 0 ) = α 3, (.) f : R R N R N y(x) = [y (x), y (x),...,y N (x)] f(x,y) = [f (x, y), f (x,y),...,f N (x, y)] α 0 = [α 0,α0,...,α0 N ] α = [α,α,...,α N ] α = [α,α,...,α N ]. α 3 = [α 3,α3,...,α3 N ].

4 4690 Mohammed S. Mechee and Murtaza A. Kadhim If we work in high dimension, then (.) can now simplified to using the following assumption y (x) y (x) y 3 (x) z(x) =......, g(z) =... y N (x) x with the initial conditions where, z (4) (x) = g(z(x)) (.3) f (z,z,...,z N,z N+ ) f (z,z,...,z N,z N+ ) f 3 (z,z,...,z N,z N+ ) f N (z,z,...,z N,z N+ ) 0 z(x 0 ) = α 0,z (x 0 ) = α,z (x 0 ) = α,z (x 0 ) = α 3, α 0 = [α 0,α0,...,α0 N,x 0], α = [α,α,...,α N, ], α = [α,α,...,α N, 0], α 3 = [α 3,α3,...,α3 N, 0]. The solution to Equation (.) or (.3) can be obtained by reducing it to an equivalent first-order system four-times the dimension and be solved using a standard Runge-Kutta method or a multistep method. Most researchers, scientists and engineers used to solve higher order ODEs by converting the nth-order ODE into a system of first-order ODEs n-times the dimensions (see [8]). Some researchers can solve this ordinary differential equation using multistep methods. However, it would be more efficient if higher order ODEs can be directly solved using special numerical methods. For second-order ODEs, [34] and [36] have derived direct numerical methods with constant step-size while [5] has derived direct numerical methods with variable step-size for solving second-order ODEs while for third-order, [5] and [4] have derived direct integrators of Runge-Kutta type for solving special third-order ODEs with constant step-size. Moreover, [], [4] and [] have derived different orders direct integrators of Runge-Kutta type for solving special third-order ODEs with constant step-size while [3] is derived a variable step-size direct integrators for Runge-Kutta type of orders 6(5), 5(4) and 4(3) for solving third-order ODEs. Accordingly, we can use RKD methods of different orders companying with the method of lines to solve third-order PDEs (see []). [8] derived direct integration implicit variable steps method for solving higher order systems of ODEs directly. However, the the regions of stability for RKD methods have been studied by [7].,

5 Solving Special Fourth-Order Ordinary Differential Equations 469 In this paper, we are concerned with the one-step method particularly Runge-Kutta integrator for directly solving fourth-order ODEs. Accordingly, we developed the order conditions for direct Runge-Kutta methods, so that based on the order conditions RKM method can be derived. 3. Proposed RKM methods The general form of RKM method with s-stage for solving fourth-order ODE (.) can be proposed as the following: y n+ = y n + hy n + h y n + h3 y n 6 + h4 b i k i (3.4) where y n+ = y n + hy n + h y n + h3 y n+ = y n + hy n + h y n+ = y n + h i= i= i= b i k i (3.5) i= b i k i (3.6) b i k i (3.7) k = f(x n,y n ) (3.8) k i = f(x n + c i h, y n + hc i y n + h c i y n + h3 6 c3y i n + i h4 a ij k j ) (3.9) for i =, 3,...,s.The parameters of RKM method are c i,a ij,b i,b i,b i,b i for i, j =,,...,s are assumed to be real. If a ij = 0 for i j, it is an explicit method and otherwise implicit method. The RKM method can be expressed in Butcher notation using the table of coefficients as follows: c A b T b T b T b T. [3] have derived the order conditions of RKD method up to order seven for solving third-order ODEs. To obtain the order conditions, we used the Taylor series expansion approach. In this paper, using the same technique, we have derived the order conditions up to order five for solving fourth-order ODEs. j=

6 469 Mohammed S. Mechee and Murtaza A. Kadhim 3.. Derivation of the order conditions of RKM methods This approach has a distinguished history as it deals with non-scalar problems with the hope that this idea may generalize correctly to high dimensions. It is similar to those which were used by Runge, Heun, Kutta, Nystrom, Huta, Mechee, You and others to obtain methods up to order six for the first-order initial value problem y = f(x,y) (see [3]). General order conditions for the RKM method can be found from the direct expansion of the local truncation error. This idea is based on the derivation of order conditions for Runge-Kutta method introduced in [7]. The RKM formulae in ( ) may be expressed as y n+ = y n + h (x n,y n ), y n+ = y n + h (x n,y n ), y n+ = y n + h (x n,y n ), y n+ = y n + h (x n,y n ), where the increment functions are (x n,y n ) = y n + h y n + h y n 6 + h3 b i k i, (x n,y n ) = y n + h y n + h (x n,y n ) = y n + h b i k i, (x n,y n ) = i= i= b i k i, b i k i, i= i= and k i is defined in formula (3.8) and (3.9). If is the Taylor series increment function, then the local truncation errors of the solution, the derivatives may be obtained by substituting the true solution y(x) of Equation (.) into the RKM increment functions. This gives t n+ = h[ ], t n+ = h[ ], t n+ = h[ ], t n+ = h[ ]. These expressions are best given in terms of elementary differentials and the Taylor series increment may be written as =y + h y + h 6 y(3) + h3 4 F (4) + O(h 4 ), =y + h y(3) + h 6 F (4) + h3 4 F (5) + O(h 4 ),

7 Solving Special Fourth-Order Ordinary Differential Equations 4693 =y (3) + h F (4) + h 6 F (5) + h3 4 F (6) + O(h 4 ), =F (4) + h F (5) + h 6 F (6) + h3 4 F (7) + O(h 4 ), where, for the scalar case the first few elementary differentials are F (4) = f, F (5) = f x + f y y, F (6) = f xx + f xy y + f y y + f yy (y ), F (7) = f xxx + y 3 f yyy + f xyy (y + y ) + 3y y f yy + 3y f xy + 3y f xyx + y f y, F (5) = f y (f x + f y y + f y f), F (5) 3 = ff y. Using the above terms the increment function,, and for the RKM formula becomes b i k i = i= b i f + i= b i c i (f x + f y y )h+ i= b i ci (f xx + f xy y + f y y + f yy (y ) )h + O(h 3 ), (3.0) i= b i k i = b i f + b i c i(f x + f y y )h+ i= i= i= b i k i = b i k i = i= i= b i c i (f xx + f xy y + f y y + f yy (y ) )h + O(h 3 ), (3.) i= i= b i f + i= i= i= b i c i(f x + f y y )h+ b i c i (f xx + f xy y + f y y + f yy (y ) )h + O(h 3 ) (3.) b i f + i= i= b i c i(f x + f y y )h+ b i c i (f xx + f xy y + f y y + f yy (y ) )h + O(h 3 ). (3.3)

8 4694 Mohammed S. Mechee and Murtaza A. Kadhim The expressions for the local truncation errors in the solution, the first derivative, the second derivative and the third derivative y,y,y and y resp. are [ ( t n+ = h 4 b i k i 6 F (4) + + )] 4 F (5), (3.4) i= [ ( t n+ = h3 b i k i F (4) + + )] 6 F (5), (3.5) i= [ ( t n+ = h b i k i F (4) + F (5) + + )] 6 h F (5), (3.6) i= [ ( t n+ = h b i k i F (4) + F (5) + + )] 6 h F (6). (3.7) i= Substituting Equations ( ) into Equations ( ) respectively and expanding as a Taylor expansion using MAPLE software as introduced by [9], then error equations or the order conditions for y,y,y and y up to order-seven in rooted trees form for RKM methods can be written in the Tables ( 4) which have all indices are from to s. Table : Rooted trees of order conditions for y 4. Derivation of RKM Methods To derive RKM methods of orders four, five and six, we use algebraic conditions up to order of the method for y,y,y and y in Tables (-4), respectively. The result is a system of nonlinear equations with unknowns which are the coefficients of the method. Three-stage fourth-order, three-stage fifth-order and four-stage sixthorder direct RKM integrators have derived and the Butcher tableaus of these integrators are shown in the Tables (6, 7) and (8) respectively. Table (5) and Figure () present a brief comparison based on the number of stages versus order for RK, RKN, RKD and the recently proposed RKM methods.

9 Solving Special Fourth-Order Ordinary Differential Equations 4695 Table : Rooted trees of order conditions for y Table 3: Rooted trees of order conditions for y 5. Stability of the RKM Methods 5.. Zero Stability of the RKM Methods Zero stability of the RKM methods is one of the criteria for convergence of the method. Zero stability is an important tool for proving stability and convergence of linear multistep methods. [7] and [3] provide more information about zero stability. Zero stability was discussed in [], where it is used to determine an upper bound on the order of convergence of linear multistep methods. In studying the zero stability of RKM methods,

10 4696 Mohammed S. Mechee and Murtaza A. Kadhim Table 4: Rooted trees of order conditions for y Table 5: Number of stages versus order for RK, RKN, RKD and the recently proposed RKM methods No. of StagesOrder No. of Stage of RK method No. of Stage of RKN method No. of Stage of RKD method No. of Stage of RKM method we can write the method as follows, ( ) y n hy n h y n+ = h 3 y n y n+ hy n h y n+ h 3 y n+,

11 Solving Special Fourth-Order Ordinary Differential Equations 4697 Table 6: The Butcher Tableau RKM4 Method p(ξ) = det[iξ A] = ξ 6 0 ξ 0 0 ξ ξ. Thus, the characteristic polynomial is, p(ξ) = (ξ ) 4. (5.8) Hence, the method is zero-stable because the roots are less than or equal to one (ξ =,,, ). 5.. Absolute Stability of The Methods In studying the linear stability of the method, we apply the test equation y = λ 4 y (see [30]), which was used as the test equation for RKN method.

12 4698 Mohammed S. Mechee and Murtaza A. Kadhim Minimum Stage Number RK RKN RKM 3.5 (a) Order of the Method 8000 No. of Functions Call RK4 RK5 RK6 RKM4 RKM5 RKM6 000 (b) Method Figure : (a) Minimum stage number versus method order for RK, RKN, and the recently proposed RKM method. (b) Number of function calls for RK and RKM methods with orders 4,5,6. We consider Formulas (3.4)-(3.7), which can be written as follows: y n+ = y n + hy n + h y n + h3 y n + h4 b i ( λ 4 Y i ), (5.9) y n+ = y n + hy n + h y n + h3 y n+ = y n + hy n + h y n+ = y n + h i= i= i= b i ( λ4 Y i ), (5.0) i= b i ( λ4 Y i ), (5.) b i ( λ4 Y i ), (5.)

13 Solving Special Fourth-Order Ordinary Differential Equations 4699 where, Y i = y n + c i hy n + c i h y n + i h3 a ij ( λ 3 Y j ), for i =,, 3,...,s. Multiplying Equations (5.0), (5.) and (5.) by h, h and h 3 respectively we obtain, where, j= y n+ = y n + hy n + h y n + h3 y n + h4 y n+ = y n + hy n + h y n + h3 y n+ = y n + hy n + h y n+ = y n + h i= i= b i ( λ 4 Y i ), (5.3) i= b i ( λ4 Y i ), (5.4) i= b i ( λ4 Y i ), (5.5) b i ( λ4 Y i ), (5.6) Y i = y n + hc i y n + h c i y n + h3 λ a ji Y j, (5.7) i j= for i =,,...,s. We can write Equations (5.3) (5.6) in the following matrix notation: z n+ = z n + λh 4 b b... b s b b... b s b b... b s b b... b s Y Y. Y s, where, z n = y n hy n h y n h 3 y n,

14 4700 Mohammed S. Mechee and Murtaza A. Kadhim and the Equation (5.7) as, c c 0 Y Y c3 Y 3 c 3 0 =.... Y cs s c s 0 cs c s 0 where, Y Y. Y s a z n + H... a s a s a s3... a ss = (I HA) CZ, Y Y Y 3. Y s, and H = λ 4 h 4 = (λh) 4. Hence, where, D(H) = z n+ = D(H)z n, + Hb T N e + Hb T N c + HbT N d Hb T N e + Hb T N c + Hb T N d, (5.8) Hb T N e Hb T N c + Hb T N d and e = (,,...,) T, c = (0,c,c 3,...,c s ) T, d = (0, c, c 3,...,c s )T, N = I HA, a A = a 3 a ,B =... a s a s a s3... a ss b b... b s b b... b s b b... b s b b... b s,

15 Solving Special Fourth-Order Ordinary Differential Equations 470 and C = 0 0 c c... c s c s. The stability function associated with this method is given by, ϕ(ξ,h) = ξi D(H), where D(H) defined in (5.8) is a stability matrix and its characteristic equation can be written as, φ(ξ,h) = P 0 (H )ξ 4 + P (H )ξ 3 + p (H )ξ + P 3 (H )ξ + P 4 (H ). 6. Implementation (Numerical Results) In this section, a set of fourth-order ODEs is solved by using the fourth, fifth and sixth -order RKM methods. Then, the same set of problems is reduced to a first-order ODEs system and solved using existing RK methods of the same order, four, five and six respectively. The numerical results are compared in Figures ( 4) to indicate the log of maximum absolute errors against the log of total time. The notations that were used are as follows: Step: Stepsize used. RKM4: The direct RKM method of fourth-order. RKM5: The direct RKM method of fifth-order. RKM6: The direct RKM method of sixth-order. RK4: Existing RK method of fourth-order. RK5: Existing RK method of fifth-order. RK6: Existing RK method of sixth-order as given by [7]. Total Time / Time : The total time in seconds to solve the problems. Max Error: Max y(x n ) y n is maximum of absolute errors of the true solutions and the computed solutions.

16 470 Mohammed S. Mechee and Murtaza A. Kadhim Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 4 6 (a) Log(Time) Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 4 (b) Log(Time) Figure : Errors versus Computational Time for RKM(4), RKM(5), RKM(6), RK4, RK5 and RK6 Methods in Problems (a) and (b)

17 Solving Special Fourth-Order Ordinary Differential Equations 4703 Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 4 (a) Log(Time) Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 4 (b) Log(Time) Figure 3: Errors versus Computational Time for RKM(4), RKM(5), RKM(6), RK4, RK5 and RK6 Methods in Problems (a) 3 and (b) 4

18 4704 Mohammed S. Mechee and Murtaza A. Kadhim 6.. Problems Tested Problem (Linear) Initial conditions, Exact solution: y(t) = sin(t). Problem (Non constant coefficients) Initial conditions, y (4) = y(t), 0 <t. y(0) = 0,y (0) =,y (0) = 0,y (0) =. y (4) = (6x 4 48x + )y(t), 0 <t b. Exact solution: y(t) = e t, b =. Problem 3 (Non linear) Initial conditions, Exact solution: y(t) = + t. Problem 4 (Non homogenous) Initial conditions, Exact solution: y(t) = ln( + t). y(0) =,y (0) = 0,y (0) =,y (0) = 0. y (4) (t) = 4y 5 (t), 0 <t. y(0) =,y (0) =,y (0) =,y (0) = 6. y (4) (t) = 6, 0 <t. ( + t) 4 y(0) = 0,y (0) =,y (0) =,y (0) =. Problem 5 ((Linear with relatively long interval)) y (4) (t) = 0.000y(t), 0 <t 0.

19 Solving Special Fourth-Order Ordinary Differential Equations 4705 Initial conditions, Exact solution: y(t) = e t 0. Problem 6 (Linear System) Initial conditions: y(0) =,y (0) = 0.,y (0) = 0.0,y (0) = y (4) (t) = y (t), y (4) (t) = 6y (t) 5y (t), y (4) 3 (t) = 8y 3(t) 65y (t) 5y (t). y (0) =, y (0) =, y y (0) =, y (0) = 3, y y 3 (0) = 3, y 3 (0) = 6, y 3 (0) =, y (0) =, (0) = 5, y (0) = 9, (0) = 4, y 3 The system is integrated over the interval [0, ]. Exact solution: y (t) = e t, y (t) = e t + e t, y 3 (t) = e t + e t + e 3t (0) = Application This subsection discuss an application of RKM method to solve the mathematical model of beam s deflection including the following fourth-order ordinary differential equation with the following initial conditions y (4) (x, y) = f(x,y); 0 <x<, (6.9) y = y = y = y = 0 at x = 0. (6.30) This equation used to extract the deformation and from the extraction forces acting on the beam induced by the load such as the displacement, rotation, moment and shear. 6.. Case Study The beam s problem on an elastic foundation is important in both the civil and mechanical, engineering fields, since it consist of a workable idealization for many problems and this problem is very often encountered in the analysis of highway, geotechnical, building,

20 4706 Mohammed S. Mechee and Murtaza A. Kadhim and railways. Its solution demands the modeling of, firstly, the mechanical behavior of the beam, secondly, the mechanical behavior of the soil as elastic sub grade and lastly, the form of relations between the beam and the soil. This problem has been popular in scientific and engineering literature ever since [43] presented his solution for the analysis of railways in 888. This solution was based on Winkler s [4] assumption that the deformation or deflection at either point is proportional to the foundation pressure at that point, and does not depend on the stress at any other point of the foundation (see Figure (5a) however, Figure (5b) shows examples of elastic foundation. S. P. Timoshenko [37] was the first to use the solution in this country when he found the strength of rails. Westergaard [4] used the solution to explain cracking in concrete pavements. Hetenyi [3] has done a great job in his famous book, Beams on Elastic Foundation, on the theoretical improvement of equations for various boundary conditions. To explain the fact that the rate of change of the deflection is a function of time for a beam resting on a viscoelastic medium, which is defined as a material whose force-deflection relations are functions of time, Freudenthal and Lorsch [4], put a velocity term τ y into the classical t equation EI 4 y + Ky = p, (6.3) x4 This ordinary differential equation was developed to include more variables. The analytical solution to these equations be difficult, especially when changing boundary conditions, so we will use the RKM method and compare the results with the results of the analytical solution. 6.. The Differential Equation of the Elastic beam Partial and ordinary differential equation are considered one of the most important topics in pure and applied mathematics. It is the connector between mathematical and engineering sciences. Topics of electrical, civil engineering are not free of some types of differential equations. Of this task in structural engineering equations are equations deformity which increased the difficulty of the solution with increasing degree of differential equation, which depends on the order of the problem in identifying as illustrated in the Equation (6.3), but There are no general mathematical methods for solving this equation, but there are some other methods that can be generalized to specific group of differential equations. Even numerical and finite elements method are not general to solve all the problems in differential equations in all conditions. The finite elements method is considered a numerical way to solve group of Partial and ordinary differential equations. It depends on dividing the given aspect into several parts or elements. One can conclude the continuous aspect behavior which is controlled by group of Partial and ordinary differential equations that are collected from the direct collection of the numerical solutions of the elements or parts that formulate it. This sectio discuss new numerical directly method to solve beam s deflection equation in the fourth order called RKM method.

21 Solving Special Fourth-Order Ordinary Differential Equations 4707 Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 8 (a) Log(Time) Log(Max Error) RK4 RK5 RK6 RKM4 RKM5 RKM6 4 6 (b) Log(Time) Figure 4: Errors versus Computational Time for RKM(4), RKM(5), RKM(6), RK4, RK5 and RK6 Methods in Problems (a) 5 and (b) 6

22 4708 Mohammed S. Mechee and Murtaza A. Kadhim (a) (b) Figure 5: (a) Deformation does not depend on the stress at any other point of the foundation (b) Examples of elastic foundation

23 Solving Special Fourth-Order Ordinary Differential Equations Mathematical Description of the Problem and the Solution The fourth-order ordinary differential equation of beam on elastic foundation with initial conditions are required describes the relationship between the beam s deflection and the applied load is given by the following: y (4) (x) = f(x,y(x)); 0 <x< (6.3) where y(x) is displacement, y (x) is rotation, y (x) is moment, y (x) is shear and f(x,y) is the normalized distributed load. The ill-posed problem of a beam on elastic foundation as following: Where y = y = y = y = 0 at x=0. y (4) (x) + y(x) = 0; 0 <x<. (6.33) f(x,y(x)) = y(x); 6..4 Analytical and Numerical Solution of the Problem The analytical solution of beam s problem is as follow: ( y = + e x c cos x + c sin x ) ( + e x c 3 cos x + c 4 sin x ) (6.34) where c = γ(α+ γ) γ(α+ δ + ) c = βγ γα+ δ + c 3 = ( + γα) γα+ δ + c 4 = ( + γβ) γα+ δ + α = cos β = sin γ = e δ = e We have used the RKM method to compare the numerical and analytical solution of beam s problem. Also the absolute Error between the numerical & analytical solutions of this problem show in the Figure 6.

24 470 Mohammed S. Mechee and Murtaza A. Kadhim Numerical & Exact Solutions NUMERICAL SOLUTION EXACT SOLUTION (a) x Absolute Errors (b) x Figure 6: (a) Numerical and Analytical Solutions of Beam s Problem (b) Absolute Errors between The Numerical & Analytical Solutions of the problem

25 Solving Special Fourth-Order Ordinary Differential Equations 47 Table 7: The Butcher Tableau RKM5 Method Discussion and Conclusion In this paper, we have derived the order conditions for direct integrators of Runge-Kutta type for special fourth-order ordinary differential equations; it has named by RKM methods. Our approach is based on Taylor series expansion. The objectives of this work are to establish direct explicit integrators of Runge-Kutta type for solving special fourth-order ordinary differential equations (ODEs). For this purpose, we generalized the integrators of Runge-Kutta type for solving special first, second and third-order ODEs (RK, RKN & RKD methods). We have derived three stage fourth-order, three stage fifth-order and four stage sixth-order RKM methods. Numerical comparisons for the derived methods with existing RK methods have been introduced. Numerical results show that the new methods are as accurate as well-known existing methods; however, they are more efficient in implementation as they require less function evaluations. As such, these methods are more cost effective, in terms of computation time, than existing methods. The mathematical model of beam s deflection has been solved numerically using RKM and then, compared with analytical solution of the model. Appendix In the following tables we define: t: a typical tree,

26 47 Mohammed S. Mechee and Murtaza A. Kadhim Table 8: The Butcher Tableau RKM6 Method t : order of tree t, S(t): extended of tree t, γ(t): density of t, φ(t) = γ(t). Acknowledgement The author would like to thank university of Kufa for supporting this research project. References [] Alomari, A., Anakira, N. R., Bataineh, A. S., & Hashim, I. 03, Approximate solution of nonlinear system of BVP arising in fluid flo problem, Mathematical Problems in Engineering, 03. [] Boutayeb, A., & Chetouani, A. 007, A mini-review of numerical methods for high-order problemsy, International Journal of Computer Mathematics, 84,

27 Solving Special Fourth-Order Ordinary Differential Equations 473 [3] Butcher, J. C. 008, Numerical methods for ordinary differential equations (John Wiley & Sons). [4] Chang, S.-P. 965, Infinite beams on an elastic foundation. [5] Cong, N. H. 00, Explicit pseudo two-step RKN methods with stepsize control, Applied Numerical Mathematics, 38, 35. [6] Cortell, R. 993, Application of the fourth-order Runge-Kutta method for the solution of high-order general initial value problems, Computers & structures, 49, 897. [7] Dormand, J. R. 996, Numerical methods for differential equations: a computational approach, Vol. 3 (CRC Press). [8] Faires, J. D., & Burden, R. 003, Numerical Methods, Thomson Learning, Inc., Pacific Grove. [9] Gander, W., & Gruntz, D. 999, Derivation of numerical methods using computer algebra, SIAM review, 4, 577. [0] Gorbenko, A., & Popov, V. 03, Visual Landmark Selection for Mobile Robot Navigation, IAENG International Journal of Computer Science, 40, 34. [] Hairer, E., Lubich, C., & Wanner, G. 006, Geometric numerical integration: structurepreserving algorithms for ordinary differential equations, Vol. 3 (Sprnger). [] He, F., Yeung, L. F., & Brown, M. 007, Discrete-time model representation for biochemical pathway systems, IAENG International Journal of Computer Science, 34. [3] Hetenyi, M. 958, Beams on Elastic Foundation. [4] Jator, S. N. 008, Numerical integrators for fourth order initial and boundary value problems, International Journal of Pure and Applied Mathematics, 47, 563. [5] Kelesoglu, O. 04, The solution of fourth order boundary value problem arising out of the beam-column theory using Adomian decomposition method, Mathematical Problems in Engineering, 04. [6] Kuang, Y. 993, Delay differential equations: with applications in population dynamics (Academic Press). [7] Lambert, J. D. 99, Numerical methods for ordinary differential systems: the initial value problem (John Wiley & Sons, Inc.) [8] Langsung, P. T. P. T. S., Majid, Z. A., & Suleiman, M. B. 006, Direct integration implicit variable steps method for solving higher order systems of ordinary differential equations directly, Sains Malaysiana, 35, 63. [9] M. K. Jain, S. R. K. Iyengar, J. S. V. S. 977, Numerical solution of a fourth-order ordinary differential equation, Journal of Engineering Mathematics,, 373.

28 474 Mohammed S. Mechee and Murtaza A. Kadhim [0] Malek, A., & Beidokhti, R. S. 006, Numerical solution for high order differential equations using a hybrid neural network-optimization method, Applied Mathematics and Computation, 83, [] Mechee, M., Ismail, F., Hussain, Z., & Siri, Z. 04a, Direct numerical methods for solving a class of third-order partial differential equations, Applied Mathematics and Computation, 47, 663. [] Mechee, M., Ismail, F., Senu, N., & Siri. 04b, A Third-Order Direct Integrators of Runge-Kutta Type for Special Third-Order Ordinary and Delay Differential Equations, Journal of Applied Sciences,. [3] Mechee, M., Ismail, F., Senu, N., & Siri, Z. 03a, Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method, Mathematical Problems in Engineering, 03. [4] Mechee, M., Ismail, F., Siri, Z., Senu, N., & Senu. 04c, A four stage sixth-order RKD method for directly solving special third order ordinary differential equations, Life Science Journal,. [5] Mechee, M., Senu, N., Ismail, F., Nikouravan, B., & Siri, Z. 03b, A Three- Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem, Mathematical Problems in Engineering, 03. [6] Mechee, M. S. 04, Direct Integrators of Runge-Kutta Type for Special Third- Order Differential Equations with thier Applications, Thesis, ISM, University of Malaya. [7] Mechee, M. S., Hussain, Z. M., & Mohammed, H. R. 06, On the reliability and stability of direct explicit Runge-Kutta integrators, Global Journal of Pure and Applied Mathematics,, [8] Samat, F., Ismail, F., & Suleiman, M. 03, Phase fitted and amplification fitted hybrid methods for solving second order ordinary differential equations, IAENG International Journal of Applied Mathematics, 43, 99. [9] Saucez, P., Wouwer, A. V., & Schiesser, W. 998, An adaptive method of lines solution of the Korteweg-de Vries equation, Computers & Mathematics with Applications, 35, 3. [30] Senu, N. 00, Runge-Kutta-Nystrom Methods For Solving Oscillatory Problems, PhD thesis, Universiti Putra Malaysia. [3] Senu, N., Mechee, M., Ismail, F., & Siri. 04, Embedded explicit Runge-Kutta type methods for directly solving special third order differential equations, Applied Mathematics and Computation, 40, 8. [3] Senu, N., Suleiman, M., Ismail, F., & Othman, M. 0, A singly diagonally implicit Runge-Kutta-Nyström method for solving oscillatory problems, IAENG International Journal of Applied Mathematics, 4, 55.

29 Solving Special Fourth-Order Ordinary Differential Equations 475 [33] Smith, H. L. 0, An introduction to delay differential equations with applications to the life sciences, Vol. 57 (Springer). [34] Sommeijer, B. P. 993, Explicit, high-order Runge-Kutta-Nyström methods for parallel computers, Applied Numerical Mathematics, 3,. [35] Twizell, E. 988, A family of numerical methods for the solution of high-order general initial value problems, Computer methods in applied mechanics and engineering, 67, 5 6. [36] Van der Houwen, P., & Sommeijer, B. 989, Diagonally implicit Runge-Kutta- Nyström methods for oscillatory problems, SIAM Journal on Numerical Analysis, 6, 44. [37] Westergaard, H. 965, Theory of Elasticity And Plasticity. [38] White, R. E., & Subramanian, V. R. 00, Computational methods in chemical engineering with maple applications (Springer). [39] Wu, T. Y., & Liu, G. R. 00, The generalized differential quadrature rule for fourth-order differential equations, International Journal for Numerical Methods in Engineering, 50, 907. [40] WU, X.-J., WANG, Y., & Price, W. 988, Multiple resonances, responses, and parametric instabilities in offshore structures, Journal of ship research, 3, 85. [4] You, X., & Chen, Z. 03, Direct integrators of Runge.Kutta type for special thirdorder ordinary differential equations, Applied Numerical Mathematics, 74, 8. [4] Z. Wang, Y. Tang, Y. D. 008, Noise Removal Based on Fourth Order Partial Differential Equation, IEEE International Conference on Innovative Computing Information and Control. [43] Zimmermann, H. 888, Die berechnung des eisenbahn oberbaues (The calculation of railway superstructures).