# Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

Authors: Kasim Hussain; Fudziah Ismail; Norazak Senu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A Runge-Kutta type
method for directly solving special fourth-order
ordinary differential equations (ODEs) which is
denoted by RKFD method is constructed. The order
conditions of RKFD method up to order five are
derived; based on the order conditions,
three-stage fourth- and fifth-order Runge-Kutta
type methods are constructed. Zero-stability of
the RKFD method is proven. Numerical results
obtained are compared with the existing
Runge-Kutta methods in the scientific literature
after reducing the problems into a system of
first-order ODEs and solving them. Numerical
results are presented to illustrate the
robustness and competency of the new methods in
terms of accuracy and number of function
evaluations.


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## Body

## 1. Introduction

This paper deals with the numerical integration of the special fourth-order ordinary differential equations (ODEs) of the form(1)yivx=fx,y,with initial conditions (2)yx0=y0,y′x0=y0′,y′′x0=y0′′,y′′′x0=y0′′′,in which the first, second, and third derivatives do not appear explicitly. This type of problems often arises in many fields of applied science such as mechanics, astrophysics, quantum chemistry, and electronic and control engineering. The general approach for solving the higher-order ordinary differential equation (ODE) is by transforming it into an equivalent first-order system of differential equations and then applying the appropriate numerical methods to solve the resulting system (see [1–5]). However, the application of such technique takes a lot of computational time (see [6, 7]). Direct integration method is proposed to avoid such computational encumbrance and increase the efficiency of the method. Many authors have proposed several numerical methods for directly approximating the solutions for the higher-order ODEs; for example, Kayode [8] proposed zero-stable predictor-corrector methods for solving fourth-order ordinary differential equations. Majid and Suleiman [9] derived one point method to solve system of higher-order ODEs. Jain et al. [10] constructed finite difference method for solving fourth-order ODEs. Waeleh et al. [11] constructed a new block method for solving directly higher-order ODEs. Awoyemi and Idowu [12] proposed a hybrid collocations method for solving third-order ODEs. Hybrid linear multistep method with three steps to solve second-order ODEs was introduced by Jator [13], and all the methods discussed above are multistep in nature.This paper primarily aims to construct a one-step method of orders four and five to solve special fourth-order ODEs directly; these new methods are self-starting in nature. The paper is organized as follows. In Section2, the derivation of the order conditions of RKFD method is presented. In Section 3, the zero-stability of RKFD method is given. In Section 4, three-stage RKFD methods of order four and order five are constructed. In Section 5, numerical examples are given to show the effectiveness and competency of the new RKFD methods as compared with the well known Runge-Kutta methods from the scientific literature. Conclusions are given in Section 6.


## 2. Derivation of the RKFD Method

The general form of RKFD method withs-stage for directly solving special fourth-order ODEs (1) can be written as follows:(3)yn+1=yn+hyn′+h22yn′′+h36yn′′′+h4∑i=1sbiki,(4)yn+1′=yn′+h2yn′′+h22yn′′′+h3∑i=1sbi′ki,(5)yn+1′′=yn′′+hyn′′′+h2∑i=1sbi′′ki,(6)yn+1′′′=yn′′′+h∑i=1sbi′′′ki,where(7)k1=fxn,yn,(8)ki=fxn+cih,yn+hciyn′+h22ci2yn′′+h36ci3yn′′′+h4∑j=1i-1aijkj.i=2,3,…,s.All parameters bi, bi′, bi′′, bi′′′, aij, and ci of the RKFD method are used for i,j=1,2,…,s and are supposed to be real. The RKFD method is an explicit method if aij=0 for i≤j and is an implicit method if aij≠0 for i≤j. The RKFD method can be represented by Butcher tableau as follows: (9)cAbTb′Tb′′Tb′′′TTo determine the parameters of the RKFD method given in (3)–(8), the RKFD method expression is expanded using the Taylor series expansion. After performing some algebraic manipulations, this expansion is equated to the true solution that is given by the Taylor series expansion. The direct expansion of the local truncation error is used to derive the general order conditions for the RKFD method. This approach depends on the derivation of order conditions for the Runge-Kutta method proposed in Dormand [14]. The RKFD method in (3)–(6) can be written as follows:(10)yn+1=yn+hψxn,yn,yn+1′=yn′+hψ′xn,yn,yn+1′′=yn′′+hψ′′xn,yn,yn+1′′′=yn′′′+hψ′′′xn,yn,where the increment functions are(11)ψxn,yn=yn′+h2yn′′+h26yn′′′+h3∑i=1sbiki,ψ′xn,yn=yn′′+h2yn′′′+h2∑i=1sbi′ki,ψ′′xn,yn=yn′′′+h∑i=1sbi′′ki,ψ′′′xn,yn=∑i=1sbi′′′ki,where ki is given in (8).The first few elementary differentials for the scalar equation are(12)F14=yiv=f,F15=fx+fyy′,F16=fxx+2fxyy′+fyy′′+fyyy′2,F17=fxxx+3fxxyy′+3fxyyy′2+3fxyy′′+3fyyy′y′′+fyyyy′3+fyy′′′.We assume that the Taylor series increment function is Δ.The local truncation errors ofy(x), y′(x), y′′(x), and y′′′(x) can be obtained after substituting the exact solution of (1) into (11) as follows:(13)τn+1=hψ-Δ,τn+1′=hψ′-Δ′,τn+1′′=hψ′′-Δ′′,τn+1′′′=hψ′′′-Δ′′′.The Taylor series increment functions of y(x), y′(x), y′′(x), and y′′′(x) can be expressed as follows:(14)Δ=y′+12hy′′+16h2y′′′+124h3F14+1120h4F15+1720h5F16+Oh6,Δ′=y′′+12hy′′′+16h2F14+124h3F15+1120h4F16+1720h5F17+Oh6,Δ′′=y′′′+12hF14+16h2F15+124h3F16+1120h4F17+1720h5F18+Oh6,Δ′′′=F14+12hF15+16h2F16+124h3F17+1720h4F18+Oh5.Substituting (12) into (11), the increment functions ψ, ψ′, ψ′′, and ψ′′′ for RKFD method become as follows:(15)∑i=1sbiki=∑i=1sbif+∑i=1sbicifx+fyy′h+12∑i=1sbici2fxx+2fxyy′+fyy′′+fyyy′2h2+Oh3,∑i=1sbiki=∑i=1sbiF14+∑i=1sbicihF15+12∑i=1sbici2h2F16+Oh3,∑i=1sbi′ki=∑i=1sbi′f+∑i=1sbi′cifx+fyy′h+12∑i=1sbi′ci2fxx+2fxyy′+fyy′′+fyyy′2h2+Oh3,∑i=1sbi′ki=∑i=1sbi′F14+∑i=1sbi′cihF15+12∑i=1sbi′ci2h2F16+Oh3,∑i=1sbi′′ki=∑i=1sbi′′f+∑i=1sbi′′cifx+fyy′h+12∑i=1sbi′′ci2fxx+2fxyy′+fyy′′+fyyy′2h2+Oh3,∑i=1sbi′′ki=∑i=1sbi′′F14+∑i=1sbi′′cihF15+12∑i=1sbi′′ci2h2F16+Oh3,∑i=1sbi′′′ki=∑i=1sbi′′′f+∑i=1sbi′′′cifx+fyy′h+12∑i=1sbi′′′ci2fxx+2fxyy′+fyy′′+fyyy′2h2+Oh3,∑i=1sbi′′′ki=∑i=1sbi′′′F14+∑i=1sbi′′′cihF15+12∑i=1sbi′′′ci2h2F16+Oh3.Using (11) and (14), the local truncation errors (13) can be written as follows:(16)τn+1=h4∑i=1sbiki-124F14+1120hF15+⋯,τn+1′=h3∑i=1sbi′ki-16F14+124hF15+⋯,τn+1′′=h2∑i=1sbi′′ki-12F14+16hF15+⋯,τn+1′′′=h∑i=1sbi′′′ki-F14+12hF15+16h2F16+⋯.By offsetting (15) into (16) and expanding as a Taylor series expansion using computer algebra package MAPLE (see [15]), the local truncation errors or the order conditions fors-stage fifth-order RKFD method can be written as follows.The order conditions fory arefourth order:(17)∑i=1sbi=124,fifth order:(18)∑i=1sbici=1120.The order conditions for y′ arethird order:(19)∑i=1sbi′=16,fourth order:(20)∑i=1sbi′ci=124,fifth order:(21)∑i=1sbi′ci2=160.The order conditions for y′′ aresecond order:(22)∑i=1sbi′′=12,third order:(23)∑i=1sbi′′ci=16,fourth order:(24)∑i=1sbi′′ci2=112,fifth order:(25)∑i=1sbi′′ci3=120.The order conditions for y′′′ arefirst order:(26)∑i=1sbi′′′=1,second order:(27)∑i=1sbi′′′ci=12,third order:(28)∑i=1sbi′′′ci2=13,fourth order:(29)∑i=1sbi′′′ci3=14,fifth order:(30)∑i=1sbi′′′ci4=15,∑i,j=1sbi′′′aij=1120.


## 3. Zero-Stability of the RKFD Method

In this section, we discuss the convergence of the RKFD method by introducing the concept of zero-stability of the RKFD method. A good numerical method is a method in which the numerical approximation to the solution converges, and zero-stability is a significant criterion for convergence. The zero-stability concept for those numerical methods that are used for solving first- and second-order ODEs can be seen in Lambert [16], Dormand [14], and Butcher [4]. The RKFD method (3)–(8) can be expressed in the matrix form as follows:(31)1000010000100000yn+1hyn+1′h2yn+1′′h3yn+1′′′=1112160111200110001ynhyn′h2yn′′h3yn′′′,where I=1000010000100000 is the identity matrix coefficients of yn+1, hyn+1′, h2yn+1′′, and h3yn+1′′′, respectively, and A=111/21/60111/200110001 is matrix coefficients of yn, hyn′, h2yn′′, and h3yn′′′, respectively.The characteristic polynomial of the RKFD method is denoted byϱ(ζ) which can be written as follows:(32)ϱζ=Iζ-A=ζ-1-1-12-160ζ-1-1-1200ζ-1-1000ζ-1.Hence, ϱ(ζ)=ζ-14.We find that all the roots areζ=1,1,1,1. Generalizing the theorem proposed by Henrici [17] for solving special second-order ODEs, therefore, the RKFD method is zero-stable since all roots are less than or equal to the value of 1.


## 4. Construction of RKFD Methods

In this section, we proceed to construct explicit RKFD methods based on the order conditions which we have derived in Section2.


### 4.1. A Three-Stage RKFD Method of Order Four

This section will focus on the derivation of a three-stage RKFD method of order four, where we use the algebraic order conditions (17), (19)-(20), (22)–(24), and (26)–(29), respectively. The resulting system of equations consists of 10 nonlinear equations with 14 unknown variables to be solved; solving the system simultaneously yields a solution with four free parameters c3, b1′, b1, and b3 as follows: (33)c2=-3-4c3-4+6c3,b1′′′=6c32-6c3+16c3-3+4c3,b2′′′=-16-72c3+108c32-54c33108c3-150c32-27+72c33,b3′′′=136c33-48c32+18c3,b1′′=6c32-6c3+16c3-3+4c3,b2′′=--20c3-18c33+33c32+4108c3-150c32-27+72c33,b3′′=--1+c336c33-48c32+18c3,b2′=--48b1′c3+72b1′c32-2+11c3-12c32123-8c3+6c32,b3′=--4+36b1′+5c3-48b1′c3-96c3+72c32+36,b2=-b1-b3+124.Thus, these free parameters can be chosen by minimizing the local truncation error norms of the fifth-order conditions according to Dormand et al. [18]. However, we have another three free parameters a21, a31, and a32 that do not appear in fourth-order conditions but they appear in the minimization of error equations for fifth-order conditions of y′′′.The error norms and global error of fifth-order conditions are defined as follows:(34)τ52=∑i=1np+1τi52,τ′52=∑i=1np′+1τi′52,τ′′52=∑i=1np′′+1τi′′52,τ′′′52=∑i=1np′′′+1τi′′′52,τg52=∑i=1np+1τi52+∑i=1np′+1τi′52+∑i=1np′′+1τi′′52+∑i=1np′′′+1τi′′′52,where τ(5), τ′(5), τ′′(5), and τ′′′(5) are the local truncation error norms for y, y′, y′′, and y′′′, respectively, and τg(5) is the global error.Consequently, we find the error norms ofy, y′, and y′′, respectively, as follows:(35)τ52=1240-360b3+11-720c32b3+480b1c3-360b1+960b3c3-14c32-2+3c32,τ′52=1240-50c32+480b1′c32-360b1′c3+48c3-72-2+3c32,τ′′52=11203-12c3+10c322-2+3c32.Our goal is to choose the free parameters c3, b1′, b1, and b3 such that the error norms of fifth-order conditions have minimal value. By plotting the graph of τ′′(5)2 versus c3 and choosing a small value of c3 in the interval [0.7,3], we find that c3=17/20 is the optimal value which yields a minimum value for τ′′(5)2=3.787878788×10-4. Substituting the value of c3=17/20 into τ(5)2 and τ′(5)2 we get (36)τ52=14403-160b1+214b32,τ′52=11760-31+544b1′2.Also through plotting the graph of τ(5)2 against b1 and b3 in the interval [-0.1,0.5] and choosing a small value of b3, we get that b3=1/20 is the optimal value which gives b1=17/200 and τ(5)2=2.272727273×10-4.Now, utilizing the same technique where we draw the graph ofτ′(5)2 versus b1′ in the interval [-1,1], we find that b1′=1/18 is the best choice, and with this value of b1′, we get τ′(5)2=4.419191919×10-4.Therefore, the error equation of the fifth-order condition ofy′′′ is as follows:(37)τ′′′52=148021601604749085+6194971660900a212+8761174400a21a31+8761174400a21a32-199207202920a21+3097600a312+6195200a31a32-1408633600a31+3097600a322-1408633600a321/2.Consequently, the global error is(38)τg52=11440648001452377332189+55754744948100a212+78850569600a21a31+78850569600a21a32-17928648262800a21+27878400a312+55756800a31a32-12677702400a31+27878400a322-12677702400a321/2.By minimizing the error norm in (37) and global error in (38) with respect to the free parameters a21, a31, and a32, we get a21=-1/5, a31=19/125, and a32=19/125, which produces τ′′′(5)2=3.7878787879×10-4 and τg(5)2=7.3068870183×10-4. Finally, all the coefficients of three-stage fourth-order RKFD method are written in Butcher tableau and denoted by RKFD4 method as follows: (39)411-151720191251912517200-7751201182091926519264740884725681001819474081331256820005457


### 4.2. A Three-Stage RKFD Method of Order Five

In this section, a three-stage RKFD method of order five will be derived. The algebraic order conditions up to order five ((17)-(18), (19)–(21), (22)–(25), and (26)–(30)) need to be solved. The resulting system of equations consists of fifteen nonlinear equations, solving the system simultaneously which results in a solution with three free parameters b1, a21, and a31 as follows: (40)c2=35+610,c3=35-610,b1′′′=19,b2′′′=49-636,b3′′′=49+636,b1′′=19,b2′′=736-618,b3′′=736+618,b1′=118,b2′=118-648,b3′=118+648,b2=148-672+-12+62b1,b3=148+672-12+62b1,a32=-131125+166125a21-a31+12625-362500.Thus, these free parameters can be chosen by minimizing the local truncation error norms of the sixth-order conditions. The error norms and the global error of the sixth-order conditions are given by(41)τ62=∑i=1np+1τi62,τ′62=∑i=1np′+1τi′62,τ′′62=∑i=1np′′+1τi′′62,τ′′′62=∑i=1np′′′+1τi′′′62,τg62=∑i=1np+1τi62+∑i=1np′+1τi′62+∑i=1np′′+1τi′′62+∑i=1np′′′+1τi′′′62,where τ(6), τ′(6), τ′′(6), and τ′′′(6) are the local truncation error norms for y, y′, y′′, and y′′′ of the RKFD method, respectively. τg(6) is the global error. The error equation of sixth-order condition for y with respect to the free parameter b1 is as follows:(42)τ62=13600-19+1080b12.The error equation τ(6)2 has a minimum value equal to zero at b1=19/1080≈0.01759259259 which leads to b2=13/1080-116/2160 and b3=13/1080+116/2160. The truncation error norms of the sixth-order condition of y, y′, y′′, and y′′′ are calculated as follows:(43)τ62=0,τ′62=11200,τ′′62=13600139+628800a212-76800a2126-9840a21-3760a216+4261/2,τ′′′62=13600362-11760a216+103200a2126+600000a216a31-36840a21-34440a31+1206-118406a31+2100000a21a31+1330800a312+4488006a312+1498800a2121/2.Also, the global error can be written as(44)τg62=13600-15520a216+26400a2126+600000a216a31-46680a21-34440a31+511+1626-118406a31+2100000a21a31+1330800a312+4488006a312+2127600a212.Now, minimizing the error coefficients in (43) and (44) with respect to the free parameters a21, a31, we obtain a21=4059/187793 and a31=-1502/532215 which gives a32=1826/569317. Thus, the error equations for y, y′, y′′, and y′′′ are computed and given by (45)τ62=0,τ′62=8.333333333×10-4,τ′′62=1.666666668×10-3,τ′′′62=1.666666667×10-3,and global error norm is (46)τg62=2.499999999×10-3.Therefore, the parameters of the three-stage fifth-order RKFD method denoted by RKFD5 can be represented in Butcher tableau as follows:(47)35+610405918779335-610-15025322151826569317191080131080-1162160131080+1162160118118-648118+64819736-618736+6181949-63649+636


## 4.1. A Three-Stage RKFD Method of Order Four

This section will focus on the derivation of a three-stage RKFD method of order four, where we use the algebraic order conditions (17), (19)-(20), (22)–(24), and (26)–(29), respectively. The resulting system of equations consists of 10 nonlinear equations with 14 unknown variables to be solved; solving the system simultaneously yields a solution with four free parameters c3, b1′, b1, and b3 as follows: (33)c2=-3-4c3-4+6c3,b1′′′=6c32-6c3+16c3-3+4c3,b2′′′=-16-72c3+108c32-54c33108c3-150c32-27+72c33,b3′′′=136c33-48c32+18c3,b1′′=6c32-6c3+16c3-3+4c3,b2′′=--20c3-18c33+33c32+4108c3-150c32-27+72c33,b3′′=--1+c336c33-48c32+18c3,b2′=--48b1′c3+72b1′c32-2+11c3-12c32123-8c3+6c32,b3′=--4+36b1′+5c3-48b1′c3-96c3+72c32+36,b2=-b1-b3+124.Thus, these free parameters can be chosen by minimizing the local truncation error norms of the fifth-order conditions according to Dormand et al. [18]. However, we have another three free parameters a21, a31, and a32 that do not appear in fourth-order conditions but they appear in the minimization of error equations for fifth-order conditions of y′′′.The error norms and global error of fifth-order conditions are defined as follows:(34)τ52=∑i=1np+1τi52,τ′52=∑i=1np′+1τi′52,τ′′52=∑i=1np′′+1τi′′52,τ′′′52=∑i=1np′′′+1τi′′′52,τg52=∑i=1np+1τi52+∑i=1np′+1τi′52+∑i=1np′′+1τi′′52+∑i=1np′′′+1τi′′′52,where τ(5), τ′(5), τ′′(5), and τ′′′(5) are the local truncation error norms for y, y′, y′′, and y′′′, respectively, and τg(5) is the global error.Consequently, we find the error norms ofy, y′, and y′′, respectively, as follows:(35)τ52=1240-360b3+11-720c32b3+480b1c3-360b1+960b3c3-14c32-2+3c32,τ′52=1240-50c32+480b1′c32-360b1′c3+48c3-72-2+3c32,τ′′52=11203-12c3+10c322-2+3c32.Our goal is to choose the free parameters c3, b1′, b1, and b3 such that the error norms of fifth-order conditions have minimal value. By plotting the graph of τ′′(5)2 versus c3 and choosing a small value of c3 in the interval [0.7,3], we find that c3=17/20 is the optimal value which yields a minimum value for τ′′(5)2=3.787878788×10-4. Substituting the value of c3=17/20 into τ(5)2 and τ′(5)2 we get (36)τ52=14403-160b1+214b32,τ′52=11760-31+544b1′2.Also through plotting the graph of τ(5)2 against b1 and b3 in the interval [-0.1,0.5] and choosing a small value of b3, we get that b3=1/20 is the optimal value which gives b1=17/200 and τ(5)2=2.272727273×10-4.Now, utilizing the same technique where we draw the graph ofτ′(5)2 versus b1′ in the interval [-1,1], we find that b1′=1/18 is the best choice, and with this value of b1′, we get τ′(5)2=4.419191919×10-4.Therefore, the error equation of the fifth-order condition ofy′′′ is as follows:(37)τ′′′52=148021601604749085+6194971660900a212+8761174400a21a31+8761174400a21a32-199207202920a21+3097600a312+6195200a31a32-1408633600a31+3097600a322-1408633600a321/2.Consequently, the global error is(38)τg52=11440648001452377332189+55754744948100a212+78850569600a21a31+78850569600a21a32-17928648262800a21+27878400a312+55756800a31a32-12677702400a31+27878400a322-12677702400a321/2.By minimizing the error norm in (37) and global error in (38) with respect to the free parameters a21, a31, and a32, we get a21=-1/5, a31=19/125, and a32=19/125, which produces τ′′′(5)2=3.7878787879×10-4 and τg(5)2=7.3068870183×10-4. Finally, all the coefficients of three-stage fourth-order RKFD method are written in Butcher tableau and denoted by RKFD4 method as follows: (39)411-151720191251912517200-7751201182091926519264740884725681001819474081331256820005457


## 4.2. A Three-Stage RKFD Method of Order Five

In this section, a three-stage RKFD method of order five will be derived. The algebraic order conditions up to order five ((17)-(18), (19)–(21), (22)–(25), and (26)–(30)) need to be solved. The resulting system of equations consists of fifteen nonlinear equations, solving the system simultaneously which results in a solution with three free parameters b1, a21, and a31 as follows: (40)c2=35+610,c3=35-610,b1′′′=19,b2′′′=49-636,b3′′′=49+636,b1′′=19,b2′′=736-618,b3′′=736+618,b1′=118,b2′=118-648,b3′=118+648,b2=148-672+-12+62b1,b3=148+672-12+62b1,a32=-131125+166125a21-a31+12625-362500.Thus, these free parameters can be chosen by minimizing the local truncation error norms of the sixth-order conditions. The error norms and the global error of the sixth-order conditions are given by(41)τ62=∑i=1np+1τi62,τ′62=∑i=1np′+1τi′62,τ′′62=∑i=1np′′+1τi′′62,τ′′′62=∑i=1np′′′+1τi′′′62,τg62=∑i=1np+1τi62+∑i=1np′+1τi′62+∑i=1np′′+1τi′′62+∑i=1np′′′+1τi′′′62,where τ(6), τ′(6), τ′′(6), and τ′′′(6) are the local truncation error norms for y, y′, y′′, and y′′′ of the RKFD method, respectively. τg(6) is the global error. The error equation of sixth-order condition for y with respect to the free parameter b1 is as follows:(42)τ62=13600-19+1080b12.The error equation τ(6)2 has a minimum value equal to zero at b1=19/1080≈0.01759259259 which leads to b2=13/1080-116/2160 and b3=13/1080+116/2160. The truncation error norms of the sixth-order condition of y, y′, y′′, and y′′′ are calculated as follows:(43)τ62=0,τ′62=11200,τ′′62=13600139+628800a212-76800a2126-9840a21-3760a216+4261/2,τ′′′62=13600362-11760a216+103200a2126+600000a216a31-36840a21-34440a31+1206-118406a31+2100000a21a31+1330800a312+4488006a312+1498800a2121/2.Also, the global error can be written as(44)τg62=13600-15520a216+26400a2126+600000a216a31-46680a21-34440a31+511+1626-118406a31+2100000a21a31+1330800a312+4488006a312+2127600a212.Now, minimizing the error coefficients in (43) and (44) with respect to the free parameters a21, a31, we obtain a21=4059/187793 and a31=-1502/532215 which gives a32=1826/569317. Thus, the error equations for y, y′, y′′, and y′′′ are computed and given by (45)τ62=0,τ′62=8.333333333×10-4,τ′′62=1.666666668×10-3,τ′′′62=1.666666667×10-3,and global error norm is (46)τg62=2.499999999×10-3.Therefore, the parameters of the three-stage fifth-order RKFD method denoted by RKFD5 can be represented in Butcher tableau as follows:(47)35+610405918779335-610-15025322151826569317191080131080-1162160131080+1162160118118-648118+64819736-618736+6181949-63649+636


## 5. Numerical Examples

In this section, some numerical examples will be solved to show the efficiency of the new RKFD methods of order four and order five, which are denoted by RKFD4 and RKFD5, respectively. The comparison is made with the well known methods in the scientific literature. We use in the numerical comparisons the criteria based on computing the maximum error in the solution (max error =max⁡(ytn-yn)) which is equal to the maximum between absolute errors of the true solutions and the computed solutions. Figures 1–7 show the efficiency curves of Log10 (max error) against the computational effort measured by Log10 (function evaluations) required by each method. The following methods are used for comparison:(i)
RKFD5: the three-stage fifth-order RKFD method derived in this paper.(ii)
RKFD4: the three-stage fourth-order RKFD method derived in this paper.(iii)
RK5Bs6: the six-stage fifth-order Runge-Kutta method given in Butcher [4].(iv)
RK5Ns6: the six-stage fifth-order Runge-Kutta method given in Hairer [5].(v)
RK4s4: the four-stage fourth-order Runge-Kutta method given in Dormand [14].(vi)
RK3/8: the four-stage fourth-order 3/8 rule Runge-Kutta method given in Butcher [4].Figure 1
The efficiency curves for Example1 with h=0.1/2i, i=0,2,3,4.Figure 2
The efficiency curves for Example2 with h=0.1/2i, i=0,2,3,4.Figure 3
The efficiency curves for Example3 with h=0.1/2i, i=0,2,3,4.Figure 4
The efficiency curves for Example4 with h=0.1/2i, i=0,2,3,4.Figure 5
The efficiency curves for Example5 with h=0.1/2i, i=0,2,3,4.Figure 6
The efficiency curves for Example6 with h=0.1/2i, i=0,2,3,4.Figure 7
The efficiency curves for Example7 with h=0.025/2i, i=0,1,2,3.Example 1.
The homogeneous linear problem is as follows:(48)yiv=-4y,y0=0,y′0=1,y′′0=2,y′′′0=2.The exact solution is given by y(x)=exsin⁡(x). The problem is integrated in the interval 0,10.Example 2.
The nonhomogeneous nonlinear problem is as follows:(49)yiv=y2+cos2⁡x+sin⁡x-1,y0=0,y′0=1,y′′0=0,y′′′0=-1.The exact solution is given by y(x)=sin⁡(x). The problem is integrated in the interval 0,10.Example 3.
The homogeneous linear problem with nonconstant coefficients is as follows(50)yiv=16x4-48x2+12y,y0=1,y′0=0,y′′0=-2,y′′′0=0.The exact solution is given by y(x)=e-x2. The problem is integrated in the interval 0,3.Example 4.
The nonlinear problem is as follows:(51)yiv=3sin⁡y3+2sin2⁡ycos7y,y0=0,y′0=1,y′′0=0,y′′′0=1.The exact solution is given by y(x)=arcsin⁡(x). The problem is integrated in the interval 0,π/4.Example 5.
The linear system is as follows:(52)yiv=e3xu,y0=1,y′0=-1,y′′0=1,y′′′0=-1.ziv=16e-xy,z0=1,z′0=-2,z′′0=4,z′′′0=-8.wiv=81e-xz,w0=1,w′0=-3,w′′0=9,w′′′0=-27.uiv=256e-xw.u0=1,u′0=-4,u′′0=16,u′′′0=-64.The exact solution is given by(53)y=e-x,z=e-2x,w=e-3x,u=e-4x.The problem is integrated in the interval 0,2.Example 6.
The nonlinear system is as follows:(54)yiv=y+1y2+z2-1w2+u2,y0=1,y′0=0,y′′0=-1,y′′′0=0.ziv=z-1y2+z2+1w2+u2,z0=0,z′0=1,z′′0=0,z′′′0=-1.wiv=16w+1y2+z2-1w2+u2,w0=1,w′0=0,w′′0=-4,w′′′0=0.uiv=16u-1y2+z2+1w2+u2.u0=0,u′0=2,u′′0=0,u′′′0=-8.The exact solution is given by(55)y=cos⁡x,z=sin⁡x,w=cos⁡2x,u=sin2x.The problem is integrated in the interval 0,2.Example 7.
The nonlinear system is as follows:(56)yiv=z2w,y0=1,y′0=1,y′′0=1,y′′′0=1.ziv=16w2u,z0=1,z′0=2,z′′0=4,z′′′0=8.wiv=81u2y5,w0=1,w′0=3,w′′0=9,w′′′0=27.uiv=256y4.u0=1,u′0=4,u′′0=16,u′′′0=64.The exact solution is given by(57)y=ex,z=e2x,w=e3x,u=e4x.The problem is integrated in the interval 0,2.


## 6. Conclusion

This paper deals with Runge-Kutta type method denoted by RKFD method for directly solving special fourth-order ODEs of the formy(iv)(x)=f(x,y). First, we derived the order conditions for RKFD method, which were then used to construct three-stage fourth- and fifth-order RKFD methods. The methods are denoted by RKFD5 and RKFD4, respectively. We also proved that the RKFD method is zero-stable. From the numerical results, we observed that the new RKFD methods are more competent as compared with the existing Runge-Kutta methods in the scientific literature. From the numerical results, we conclude that the new RKFD methods are computationally more efficient in solving special fourth-order ODEs and outperformed the existing methods in terms of error precision and number of function evaluations.



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*Source: 893763-2015-08-06.xml*