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Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 4687–4715© Research India Publicationshttp://www.ripublication.com/gjpam.htm

Direct Explicit Integrators of RK Type forSolving 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 integratorsof Runge-Kutta type for solving special fourth-order ordinary differential equations(ODEs) which is denoted as RKM. For this purpose, we generalized the integratorsof Runge-Kutta type for solving special first, second and third-order ODEs (RK,RKN & RKD). Using Taylor expansion, we have derived the order conditions forthe proposed RKM integrators, up to the seventh order. Based on these order con-ditions, three RKM methods of orders four, five and six with different stages arederived. Numerical implementation shows that the new methods agree well withexisting RK methods but require less function evaluations. This is so due to the factthat RKM methods are direct; hence, they save considerable amount of computa-tional time. An application for RKM method has been introduced for solving themathematical model of beam’s deflection.

AMS subject classification: 05C78.Keywords: Runge-Kutta method; RK, RKN, RKD Integrators; fourth-order; Ordi-nary differential equations, Application of beam model.

4688 Mohammed S. Mechee and Murtaza A. Kadhim

1. Introduction

The most important mathematical model for physical phenomena is the differential equa-tion. Motion of objects, Fluid and heat flow, bending and cracking of materials, vibra-tions, chemical reactions and nuclear reactions are all modeled by systems of differentialequations. Moreover, Numerous mathematical models in science and engineering areexpressed in terms of unknown quantities and their derivatives. Many applications ofdifferential equations (DEs), particularly ODEs of different orders, can be found in themathematical modeling of real life problems. Fourth-order ODEs & PDEs occur in thephysics and engineering ([26], [16], [29], [38], [19], [39], [42] and [33]). In addition tothis type of Fourth-order DEs often arise in many fields of applied science such as me-chanics, quantum chemistry, electronic and control engineering. However, Fourth orderODEs arise in several fields such as fluid dynamics ([1]), beam theory ([14] and [15]),electric circuits ([2]), ship dynamics ([40], [35] and [6]), neural networks ([20]), and themathematical model of beam’s deflection ([37] & [4]). Finding the solutions to these dif-ferential equations had been challenged the ingenuity of mathematicians since the timeof Newton, thereby resulting in several powerful analytical techniques now available tomodern scientists. Therefore, many theoretical and numerical studies dealing with thesolution 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 differen-tial equations. Unfortunately analytical tools frequently are inadequate for the solutionof such systems. The number of cases where an exact solution can be found by analyticalmeans is very limited, the only general class of systems for which exact solutions canalways be found consists of linear systems with constant coefficients. However, in ap-plications it is not necessary to find the solution to more than a finite number of decimalplaces. For this reason numerical methods were developed for the solution of ordinarydifferential equations since the age of Newton, Taylor and Euler. There are also manyexamples of particular linear variable coefficient or nonlinear systems for which exactsolutions are known, but in general for such systems we must resort to either an approx-imate or a numerical method. These methods of solutions are not able to solve manytypes of differential equations or they can solved some types of differential equationsindirectly. This reason make us to study and derive more direct numerical methods forthis propose. In order to apply indirect numerical method to solve a differential equationof higher than order one, the equation should be transformed into a system of first orderdifferential equations.

Many researchers developed the family of Runge Kutta methods for solving first,second and third order ordinary differential equations, For example [32] has developeda singly diagonally implicit Runge-Kutta-Nyström (RKN) method for second-order or-dinary differential equations with periodical solutions. Many applications has beensolved base Runge Kutta methods. [12] solved discrete-time model representation forbiochemical pathway systems based on Runge–Kutta method. One advantage of thisapproach is to improves the discretization accuracy by utilizing multi-step informationfor approximation of one-step-ahead model prediction. Moreover, it gives a general ex-act discrete-time representation for both linear and nonlinear biochemical ODE models.

Solving Special Fourth-Order Ordinary Differential Equations 4689

Another application in the field of visual Landmark selection for mobile robot navigationevaluated based on Runge Kutta methods for finding optimal solutions requires to solvesome hard problems. [10] consider the minimum overlapping region decompositionproblem that was proposed for landmarks selection. He was describe an approach tosolve the problem optimally. This approach is based on an explicit reduction from theproblem to the satisfy ability problem. The mathematical problems often arise in engi-neering and applied sciences such as celestial mechanics, quantum mechanics, elasticdynamics, theoretical physics, chemistry and electronics and can be solved by usingRunge Kutta methods and multistep methods. It is important to increase the order ofthe methods to achieve higher accuracy. [28] derived some efficient methods for solvingsecond order ordinary differential equations which have oscillating solutions, it is alsoessential 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 explicitintegrators of Runge-Kutta type for solving special fourth-order ordinary differentialequations (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 RKmethods.

2. Preliminary

2.1. Special Fourth-Order Ordinary Differential Equations

There are special fourth-order ODEs with no explicit dependence on the first, secondand 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:

y(4)(x) = f (x, y(x)), x ≥ x0, (2.1)

with initial conditions,

y(x0) = α0, y′(x0) = α1, y

′′(x0) = α2 and y

′′′(x0) = α3, (2.2)

where

f : R × RN → R

N

and

y(x) = [y1(x), y2(x), . . . , yN(x)]f (x, y) = [f1(x, y), f2(x, y), . . . , fN(x, y)]

α0 = [α01, α

02, . . . , α

0N ]

α1 = [α11, α

12, . . . , α

1N ]

α2 = [α21, α

22, . . . , α

2N ].

α3 = [α31, α

32, . . . , α

3N ].


If we work in high dimension, then (2.1) can now simplified to

z(4)(x) = g(z(x)) (2.3)

using the following assumption

z(x) =

y1(x)

y2(x)

y3(x)

. . .

. . .

. . .

yN(x)

x

, g(z) =

f1(z1, z2, . . . , zN, zN+1)

f2(z1, z2, . . . , zN, zN+1)

f3(z1, z2, . . . , zN, zN+1)

. . .

. . .

. . .

fN(z1, z2, . . . , zN, zN+1)

0

,

with the initial conditions

z(x0) = α0, z′(x0) = α1, z

′′(x0) = α2, z

′′′(x0) = α3,

where,

α0 = [α01, α

02, . . . , α

0N, x0],

α1 = [α11, α

12, . . . , α

1N, 1],

α2 = [α21, α

22, . . . , α

2N, 0],

α3 = [α31, α

32, . . . , α

3N, 0].

The solution to Equation (2.1) or (2.3) can be obtained by reducing it to an equivalentfirst-order system four-times the dimension and be solved using a standard Runge-Kuttamethod or a multistep method. Most researchers, scientists and engineers used to solvehigher order ODEs by converting the nth-order ODE into a system of first-order ODEsn-times the dimensions (see [8]).

Some researchers can solve this ordinary differential equation using multistep meth-ods. However, it would be more efficient if higher order ODEs can be directly solvedusing special numerical methods. For second-order ODEs, [34] and [36] have deriveddirect numerical methods with constant step-size while [5] has derived direct numericalmethods with variable step-size for solving second-order ODEs while for third-order,[25] and [41] have derived direct integrators of Runge-Kutta type for solving specialthird-order ODEs with constant step-size. Moreover, [21], [24] and [22] have deriveddifferent orders direct integrators of Runge-Kutta type for solving special third-orderODEs with constant step-size while [31] is derived a variable step-size direct integratorsfor Runge-Kutta type of orders 6(5), 5(4) and 4(3) for solving third-order ODEs. Ac-cordingly, we can use RKD methods of different orders companying with the methodof lines to solve third-order PDEs (see [21]). [18] derived direct integration implicitvariable steps method for solving higher order systems of ODEs directly. However, thethe regions of stability for RKD methods have been studied by [27].


In this paper, we are concerned with the one-step method particularly Runge-Kuttaintegrator for directly solving fourth-order ODEs. Accordingly, we developed the orderconditions for direct Runge-Kutta methods, so that based on the order conditions RKMmethod can be derived.

3. Proposed RKM methods

The general form of RKM method with s-stage for solving fourth-order ODE (2.1) canbe proposed as the following:

yn+1 = yn + hy′n + h2

2y

′′n + h3

6y

′′′n + h4

s∑i=1

biki (3.4)

y′n+1 = y

′n + hy

′′n + h2

2y

′′′n + h3

s∑i=1

b′iki (3.5)

y′′n+1 = y

′′n + hy

′′′n + h2

s∑i=1

b′′i ki (3.6)

y′′′n+1 = y

′′′n + h

s∑i=1

b′′′i ki (3.7)

where

k1 = f (xn, yn) (3.8)

ki = f (xn + cih, yn + hciy′n + h2

2c2i y

′′n + h3

6c3i y

′′′n + h4

i−1∑j=1

aij kj ) (3.9)

for i = 2, 3, . . . , s. The parameters of RKM method are ci, aij , bi, b′i , b

′′i , b

′′′i for i, j =

1, 2, . . . , s are assumed to be real. If aij = 0 for i ≤ j , it is an explicit method andotherwise implicit method. The RKM method can be expressed in Butcher notationusing the table of coefficients as follows:

c A

bT

b′T

b′′T

b′′′T .

[23] have derived the order conditions of RKD method up to order seven for solvingthird-order ODEs. To obtain the order conditions, we used the Taylor series expansionapproach. In this paper, using the same technique, we have derived the order conditionsup to order five for solving fourth-order ODEs.


3.1. Derivation of the order conditions of RKM methods

This approach has a distinguished history as it deals with non-scalar problems with thehope that this idea may generalize correctly to high dimensions. It is similar to thosewhich were used by Runge, Heun, Kutta, Nystrom, Huta, Mechee, You and others toobtain 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 directexpansion of the local truncation error. This idea is based on the derivation of orderconditions for Runge-Kutta method introduced in [7]. The RKM formulae in (3.4–3.7)may be expressed as

yn+1 = yn + h�(xn, yn),

y′n+1 = y

′n + h�

′(xn, yn),

y′′n+1 = y

′′n + h�

′′(xn, yn),

y′′′n+1 = y

′′′n + h�

′′′(xn, yn),

where the increment functions are

�(xn, yn) = y′n + h

2y

′′n + h2

6y

′′′n + h3

s∑i=1

biki,

�′(xn, yn) = y

′′n + h

2y

′′′n + h2

s∑i=1

b′iki,

�′′(xn, yn) = y

′′′n + h

s∑i=1

b′′i ki,

�′′′(xn, yn) =

s∑i=1

b′′′i ki,

and ki is defined in formula (3.8) and (3.9). If � is the Taylor series increment func-tion, then the local truncation errors of the solution, the derivatives may be obtained bysubstituting the true solution y(x) of Equation (2.1) into the RKM increment functions.This gives

tn+1 = h[�−�], t′n+1 = h[�′ −�

′ ], t′′n+1 = h[�′′ −�

′′ ], t′′′n+1 = h[�′′′ −�

′′′ ].These expressions are best given in terms of elementary differentials and the Taylor seriesincrement may be written as

� =y′ + h

2y

′′ + h2

6y(3) + h3

24F

(4)1 + O(h4),

�′ =y

′′ + h

2y(3) + h2

6F

(4)1 + h3

24F

(5)1 + O(h4),


�′′ =y(3) + h

2F

(4)1 + h2

6F

(5)1 + h3

24F

(6)1 + O(h4),

�′′′ =F

(4)1 + h

2F

(5)1 + h2

6F

(6)1 + h3

24F

(7)1 + O(h4),

where, for the scalar case the first few elementary differentials are

F(4)1 = f,

F(5)1 = fx + fyy

′,

F(6)1 = fxx + 2fxyy

′ + fyy′′ + fyy(y

′)2,

F(7)1 = fxxx + y

′3fyyy + fxyy(2y′ + y

′2) + 3y′y

′′fyy + 3y

′′fxy + 3y

′fxyx + y

′′′fy,

F(5)2 = fy

′ (fx + fyy′ + fy

′f ),

F(5)3 = ffy.

Using the above terms the increment function �, �′, �

′′and �

′′′for the RKM formula

becomes

s∑i=1

biki =s∑

i=1

bif +s∑

i=1

bici(fx + fyy′)h+

1

2

s∑i=1

bic2i (fxx + 2fxyy


′)2)h2 + O(h3), (3.10)

s∑i=1

b′iki =

s∑i=1

b′if +

s∑i=1

b′ici(fx + fyy

′)h+

1

2

s∑i=1

b′ic

2i (fxx + 2fxyy


′)2)h2 + O(h3), (3.11)

s∑i=1

b′′i ki =

s∑i=1

b′′i f +

s∑i=1

b′′i ci(fx + fyy

′)h+

1

2

s∑i=1

b′′i c

2i (fxx + 2fxyy


′)2)h2 + O(h3) (3.12)

s∑i=1

b′′′i ki =

s∑i=1

b′′′i f +

s∑i=1

b′′′i ci(fx + fyy

′)h+

1

2

s∑i=1

b′′′i c2

i (fxx + 2fxyy′ + fyy

′′ + fyy(y′)2)h2 + O(h3). (3.13)


The expressions for the local truncation errors in the solution, the first derivative, thesecond derivative and the third derivative y, y′, y′′ and y′′′ resp. are

tn+1 = h4

[s∑

i=1

biki −(

1

6F

(4)1 + 1

24F

(5)1 + · · ·

)], (3.14)

t′n+1 = h3

[s∑

i=1

b′iki −

(1

2F

(4)1 + 1

6F

(5)1 + · · ·

)], (3.15)

t′′n+1 = h2

[s∑

i=1

b′′i ki −

(F

(4)1 + 1

2F

(5)1 + 1

6h2F

(5)1 + · · ·

)], (3.16)

t′′′n+1 = h

[s∑

i=1

b′′′i ki −

(F

(4)1 + 1

2F

(5)1 + 1

6h2F

(6)1 + · · ·

)]. (3.17)

Substituting Equations (3.10–3.13) into Equations (3.14–3.17) respectively and expand-ing as a Taylor expansion using MAPLE software as introduced by [9], then error equa-tions or the order conditions for y, y′, y′′ and y′′′ up to order-seven in rooted trees formfor RKM methods can be written in the Tables (1–4) which have all indices are from 1to s.

Table 1: 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 toorder of the method for y, y

′, y

′′and y

′′′in Tables (1-4), respectively.

The result is a system of nonlinear equations with unknowns which are the coefficientsof the method. Three-stage fourth-order, three-stage fifth-order and four-stage sixth-order direct RKM integrators have derived and the Butcher tableaus of these integratorsare shown in the Tables (6, 7) and (8) respectively. Table (5) and Figure (1) present abrief comparison based on the number of stages versus order for RK, RKN, RKD andthe recently proposed RKM methods.


Table 2: Rooted trees of order conditions for y′

Table 3: Rooted trees of order conditions for y′′

5. Stability of the RKM Methods

5.1. 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 multistepmethods. [17] and [3] provide more information about zero stability. Zero stabilitywas discussed in [11], where it is used to determine an upper bound on the order ofconvergence of linear multistep methods. In studying the zero stability of RKM methods,


Table 4: Rooted trees of order conditions for y′′′

Table 5: Number of stages versus order for RK, RKN, RKD and the recently proposedRKM methods

.

No. of StagesOrder 3 4 5 6

No. of Stage of RK method 3 4 6 7No. of Stage of RKN method 2 3 4 5No. of Stage of RKD method 2 3 3 4No. of Stage of RKM method 2 3 4 4

we can write the method as follows, (3.4–3.7)

1 0 0 00 1 0 00 0 1 00 0 0 1

yn+1

hy′n

h2y′′n+1

h3y′′′n+1

=

1 11

2

1

60 1 1

1

20 0 1 10 0 0 1

yn+1

hy′n

h2y′′n+1

h3y′′′n+1

,


Table 6: The Butcher Tableau RKM4 Method

0 0

5

6

1

20

1

3

1

2

1

20

1

60

1

3240

2

81

1

20

1

180

1

9

1

10

1

15

1

3

1

10

2

5

1

2

p(ξ) = det[Iξ − A] =

∣∣∣∣∣∣∣∣∣∣∣

ξ − 1 −1 −1

2−1

60 ξ − 1 −1 −1

20 0 ξ − 1 −10 0 0 ξ − 1

∣∣∣∣∣∣∣∣∣∣∣.

Thus, the characteristic polynomial is,

p(ξ) = (ξ − 1)4. (5.18)

Hence, the method is zero-stable because the roots are less than or equal to one (ξ =1, 1, 1, 1).

5.2. Absolute Stability of The Methods

In studying the linear stability of the method, we apply the test equation

y′′′′ = −λ4y

(see [30]), which was used as the test equation for RKN method.


(a)4 4.5 5 5.5 6

3

3.5

4

4.5

5

5.5

6

6.5

7

Order of the Method

Min

imu

m S

tag

e N

um

ber

RK RKN RKM

(b)1 2 3 4 5 6

0

1000

2000

3000

4000

5000

6000

7000

8000

RK4

RK5

RK6

RKM4 RKM5

RKM6

Method

No

. of

Fu

nct

ion

s C

all

Figure 1: (a) Minimum stage number versus method order for RK, RKN, and the recentlyproposed RKM method. (b) Number of function calls for RK and RKM methods withorders 4,5,6.

We consider Formulas (3.4)-(3.7), which can be written as follows:

yn+1 = yn + hy′n + h2

2y

′′n + h3

2y

′′′n + h4

s∑i=1

bi(−λ4Yi), (5.19)

y′n+1 = y

′n + hy

′′n + h2

2y

′′n + h3

s∑i=1

b′i(−λ4Yi), (5.20)

y′′n+1 = y

′′n + hy

′′′n + h2

s∑i=1

b′′i (−λ4Yi), (5.21)

y′′′n+1 = y

′′′n + h

s∑i=1

b′′′i (−λ4Yi), (5.22)


where,

Yi = yn + cihy′n + c2

i

2h2y

′′n + h3

i−1∑j=1

aij (−λ3Yj ),

for i = 1, 2, 3, . . . , s.Multiplying Equations (5.20), (5.21) and (5.22) by h, h2 and h3 respectively we

obtain,

yn+1 = yn + hy′n + h2

2y

′′n + h3

2y

′′′n + h4

s∑i=1

bi(−λ4Yi), (5.23)

y′n+1 = y

′n + hy

′′n + h2

2y

′′n + h3

s∑i=1

b′i(−λ4Yi), (5.24)

y′′n+1 = y

′′n + hy

′′′n + h2

s∑i=1

b′′i (−λ4Yi), (5.25)

y′′′n+1 = y

′′′n + h

s∑i=1

b′′′i (−λ4Yi), (5.26)

where,

Yi = yn + hciy′n + h2 c2

i

2y

′′n + h3λ

i−1∑j=1

ajiYj , (5.27)

for i = 1, 2, . . . , s.

We can write Equations (5.23)–(5.26) in the following matrix notation:

zn+1 =

1 11

2

1

60 1 1

1

20 0 1 10 0 1 1

zn + λh4

b1 b2 . . . bs

b′1 b

′2 . . . b

′s

b′′1 b

′′2 . . . b

′′s

b′′′1 b

′′′2 . . . b

′′′s

Y1

Y2...

Ys

,

where,

zn =

yn

hy′n

h2y′′n

h3y′′′n

,


and the Equation (5.27) as,

Y1

Y2

Y3...

Ys

=

1 0 0 0

1 c2 0c2

2

2

1 c3 0c2

3

2. . .

1 cs

c2s

20

1 cs

c2s

20

zn + H

0 0 0 . . . 0a21 0 0 . . . 0

. . .

as1 as2 as3 . . . ass

Y1

Y2

Y3...

Ys

,

where,

Y1

Y2...

Ys

= (I − HA)−1 CZ,

and

H = λ4h4 = (λh)4.

Hence,

zn+1 = D(H)zn,

where,

D(H) =

1 + HbT N−1e 1 + HbT N−1c1

2+ HbT N−1d

Hb′T N−1e 1 + Hb

′T N−1c 1 + Hb′T N−1d

Hb′′T N−1e Hb

′′T N−1c 1 + Hb′′T N−1d

, (5.28)

e = (1, 1, . . . , 1)T , c = (0, c2, c3, . . . , cs)T ,

and

d = (0,c2

2

2,c2

3

2, . . . ,

c2s

2)T , N−1 = I − HA,

A =

0 0 . . . 0a21 0 . . . 0a31 a32 0 . . . 0

. . .

as1 as2 as3 . . . ass

, B =

b1 b2 . . . bs

b′1 b

′2 . . . b

′s

b′′1 b

′′2 . . . b

′′s

b′′′1 b

′′′2 . . . b

′′′s

,


and

C =

1 0 0

1 c2c2

2

2. . .

1 cs

c2s

2

.

The stability function associated with this method is given by,

ϕ(ξ, H) = |ξI − D(H)|,where D(H) defined in (5.28) is a stability matrix and its characteristic equation can bewritten as,

φ(ξ, H) = P0(H)ξ4 + P1(H)ξ3 + p2(H)ξ2 + P3(H)ξ + P4(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 ODEssystem and solved using existing RK methods of the same order, four, five and sixrespectively. The numerical results are compared in Figures (2–4) to indicate the log ofmaximum absolute errors against the log of total time. The notations that were used areas 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(xn)− yn| is maximum of absolute errors of the true solutionsand the computed solutions.


(a)

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−18

−16

−14

−12

−10

−8

−6

−4

−2

Log(Time)

L

og

(Ma

x E

rro

r)

RK4 RK5 RK6 RKM4 RKM5 RKM6

(b)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−16

−14

−12

−10

−8

−6

−4

−2

Log(Time)

L

og

(Ma

x E

rro

r)


Figure 2: Errors versus Computational Time for RKM(4), RKM(5), RKM(6), RK4, RK5and RK6 Methods in Problems (a) 1 and (b) 2


(a)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−16

−14

−12

−10

−8

−6

−4

−2

Log(Time)

L

og

(Ma

x E

rro

r)


(b)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−16

−14

−12

−10

−8

−6

−4

−2

Log(Time)

L

og

(Ma

x E

rro

r)




6.1. Problems Tested

Problem 1 (Linear)

y(4) = y(t), 0 < t ≤ 1.

Initial conditions,

y(0) = 0, y′(0) = 1, y

′′(0) = 0, y

′′′(0) = −1.

Exact solution: y(t) = sin(t).

Problem 2 (Non constant coefficients)

y(4) = (16x4 − 48x2 + 12)y(t), 0 < t ≤ b.

Initial conditions,

y(0) = 1, y′(0) = 0, y

′′(0) = −2, y

′′′(0) = 0.

Exact solution: y(t) = e−t2, b = 1.

Problem 3 (Non linear)

y(4)(t) = 24y5(t), 0 < t ≤ 1.

Initial conditions,

y(0) = 1, y′(0) = −1, y

′′(0) = 2, y

′′′(0) = −6.

Exact solution: y(t) = 1

1 + t.

Problem 4 (Non homogenous)

y(4)(t) = − 6

(1 + t)4, 0 < t ≤ 1.

Initial conditions,

y(0) = 0, y′(0) = 1, y

′′(0) = −1, y

′′′(0) = 2.

Exact solution: y(t) = ln(1 + t).

Problem 5 ((Linear with relatively long interval))

y(4)(t) = 0.0001y(t), 0 < t ≤ 10.


Initial conditions,

y(0) = 1, y′(0) = −0.1, y

′′(0) = 0.01, y

′′′(0) = −0.001.

Exact solution: y(t) = e− t10 .

Problem 6 (Linear System)

y(4)1 (t) = y1(t),

y(4)2 (t) = 16y2(t) − 15y1(t),

y(4)3 (t) = 81y3(t) − 65y2(t) − 15y1(t).

Initial conditions:

y1(0) = 1, y′1(0) = −1, y

′′1(0) = 1, y

′′′1 (0) = −1,

y2(0) = 2, y′2(0) = −3, y

′′2(0) = 5, y

′′′2 (0) = −9,

y3(0) = 3, y′3(0) = −6, y

′′3(0) = 14, y

′′′3 (0) = −36.

The system is integrated over the interval [0, 1]. Exact solution:

y1(t) = e−t ,

y2(t) = e−t + e−2t ,

y3(t) = e−t + e−2t + e−3t

6.2. Application

This subsection discuss an application of RKM method to solve the mathematical modelof beam’s deflection including the following fourth-order ordinary differential equation

y(4)(x, y) = f (x, y); 0 < x < 1, (6.29)

with the following initial conditions

y = y′ = y

′′ = y′′′ = 0 at x = 0. (6.30)

This equation used to extract the deformation and from the extraction forces acting onthe beam induced by the load such as the displacement, rotation, moment and shear.

6.2.1 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 thisproblem is very often encountered in the analysis of highway, geotechnical, building,


and railways. Its solution demands the modeling of, firstly, the mechanical behavior ofthe 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 popularin scientific and engineering literature ever since [43] presented his solution for theanalysis of railways in 1888. This solution was based on Winkler’s [4] assumption thatthe deformation or deflection at either point is proportional to the foundation pressureat that point, and does not depend on the stress at any other point of the foundation (seeFigure (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[13] has done a great job in his famous book, Beams on Elastic Foundation, on thetheoretical improvement of equations for various boundary conditions. To explain thefact that the rate of change of the deflection is a function of time for a beam resting on aviscoelastic 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

∂tinto the classical

equation

EI∂4y

∂x4+ Ky = p, (6.31)

This ordinary differential equation was developed to include more variables. The an-alytical solution to these equations be difficult, especially when changing boundaryconditions, so we will use the RKM method and compare the results with the results ofthe analytical solution.

6.2.2 The Differential Equation of the Elastic beam

Partial and ordinary differential equation are considered one of the most important top-ics in pure and applied mathematics. It is the connector between mathematical andengineering sciences. Topics of electrical, civil engineering are not free of some typesof differential equations. Of this task in structural engineering equations are equationsdeformity which increased the difficulty of the solution with increasing degree of differ-ential equation, which depends on the order of the problem in identifying as illustratedin the Equation (6.31), but There are no general mathematical methods for solving thisequation, but there are some other methods that can be generalized to specific groupof differential equations. Even numerical and finite elements method are not generalto solve all the problems in differential equations in all conditions. The finite elementsmethod is considered a numerical way to solve group of Partial and ordinary differentialequations. It depends on dividing the given aspect into several parts or elements. Onecan conclude the continuous aspect behavior which is controlled by group of Partialand ordinary differential equations that are collected from the direct collection of thenumerical solutions of the elements or parts that formulate it. This sectio discuss newnumerical directly method to solve beam’s deflection equation in the fourth order calledRKM method.


(a)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−20

−18

−16

−14

−12

−10

−8

−6

Log(Time)

L

og

(Ma

x E

rro

r)


(b)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−18

−16

−14

−12

−10

−8

−6

−4

−2

Log(Time)

L

og

(Ma

x E

rro

r)




(a)

(b)

Figure 5: (a) Deformation does not depend on the stress at any other point of the foun-dation (b) Examples of elastic foundation


6.2.3 Mathematical Description of the Problem and the Solution

The fourth-order ordinary differential equation of beam on elastic foundation with initialconditions are required describes the relationship between the beam’s deflection and theapplied load is given by the following:

y(4)(x) = f (x, y(x)); 0 < x < 1 (6.32)

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 foundationas following:

y(4)(x) + y(x) − 1 = 0; 0 < x < 1. (6.33)

f (x, y(x)) = 1 − y(x);Where y = y

′ = y′′ = y

′′′ = 0 at x=0.

6.2.4 Analytical and Numerical Solution of the Problem

The analytical solution of beam’s problem is as follow:

y = 1 + ex√2

(c1 cos

x√2

+ c2 sinx√2

)+ e

− x√2

(c3 cos

x√2

+ c4 sinx√2

)(6.34)

where

c1 = 1 − γ (α + γ )

2γ (α + δ + 1)

c2 = − βγ

2γα + δ + 1

c3 = − (1 + γα)

2γα + δ + 1

c4 = − (1 + γβ)

2γα + δ + 1

α = cos1√2

β = sin1√2

γ = e− 1√

2

δ = e−√2

We have used the RKM method to compare the numerical and analytical solution ofbeam’s problem. Also the absolute Error between the numerical & analytical solutionsof this problem show in the Figure 6.


(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x

N

um

eri

ca

l &

Ex

ac

t S

olu

tio

ns

NUMERICAL SOLUTION EXACT SOLUTION

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4−13

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

x

A

bs

olu

te E

rro

rs

Figure 6: (a) Numerical and Analytical Solutions of Beam’s Problem (b) Absolute Errorsbetween The Numerical & Analytical Solutions of the problem



0 0

3

5−

√6

10−643

625+ 317

2500

√6 0

3

5+

√6

10

1

2

1

20

1

54

5

432+

√6

216

5

432−

√6

216

1

18

1

18+

√6

48

1

18−

√6

48

1

9

10915

33023

3397

58206

1

9

4

9+

√6

36

4

9−

√6

36

7. Discussion and Conclusion

In this paper, we have derived the order conditions for direct integrators of Runge-Kuttatype for special fourth-order ordinary differential equations; it has named by RKMmethods. Our approach is based on Taylor series expansion. The objectives of thiswork are to establish direct explicit integrators of Runge-Kutta type for solving specialfourth-order ordinary differential equations (ODEs). For this purpose, we generalized theintegrators 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 stagefifth-order and four stage sixth-order RKM methods. Numerical comparisons for thederived methods with existing RK methods have been introduced. Numerical resultsshow 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. Assuch, these methods are more cost effective, in terms of computation time, than existingmethods. The mathematical model of beam’s deflection has been solved numericallyusing RKM and then, compared with analytical solution of the model.

Appendix

In the following tables we define:

• t: a typical tree,



0 01

23

160−

√15

240

1

2−

√15

10

1

2−1

20

1

2+

√15

10− 51

100− 22

75

√15

1

100+

√15

5

1

20

01

108

7

432+

√15

240

7

432−

√15

240

01

18

1

18+

√15

72

1

18−

√15

72

02

9

5

36+

√15

36

5

36−

√15

36

04

9

5

18

5

18

• |t|: order of tree t ,

• S(t): extended of tree t ,

• γ (t): density of t ,

• φ(t) = 1γ (t)

.

Acknowledgement

The author would like to thank university of Kufa for supporting this research project.

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