J. Appl. Environ. Biol. Sci.,8(7)26-35,2018 | ISSN:2090-4274 |
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MuhammadDaudKandhro1,AsifAliShaikh2,Sania Qureshi3
Department ofBasic Sciences and Related Studies, Mehran UniversityofEngineeringandTechnology, Jamshoro, Pakistan
Received:January14,2018 Accepted:April1,2018
ABSTRACT
In this paper, an explicit and single step Runge-Kutta iterative integrator of third order has been developed which is used to solve both autonomous and non-autonomous type of initial value problems also called Cauchy problems in ordinary differential equations. Linear stability analysis with corresponding stability region is drawn and error analysis has been provided to confirm third order accuracy of the integrator. Inclusion of a partial derivative with respect to the dependent variable within two slopes of the integrator has improved its efficiency in terms of local and global truncation errors. Finally, numerical examples are provided to show performance of the proposed integrator in comparison with other existing methods having same order of local accuracy. The software MATLAB R2017b was employed in order to produce all the numerical results and graphicalillustrations presented in this paper whereas theMATLABcodedesigned toget such numerical results for all theintegrators under considerationhave alsobeen provided. KEYWORDS:Runge-Kutta, Stability, Local Truncation Error, Non-autonomous, Cauchyproblems.
Ordinarydifferential equations usuallyexpress all natural phenomena we come across in this physical universe. Their usage is present everywhere in biology, environmental engineering, physical systems, business, and economics toname a few [1-3]. It is acommon practice ofphysical and biological researchers around the world to study the mathematical modeling of various physical problems based upon Cauchy problems such as radioactive decay, population dynamics, mechanical systems, fluid flows, electrical networks, rate of chemical reactions and manymore [4-6]. Manyproblems of mathematical physics can be represented in the form of such ordinarydifferential equations. In manysituations, the methods capable to obtain solution of certain mathematical models stand to be veryhard and complicated and sometimes even fail to produce the required results [7-9]. There are various numerical methods to get approximate solution of an initial value problem in ordinary differential equations and this is because only one numerical method cannot serve the purpose to get the solution of every type of initial value problem. Specially, numerous nonlinear types of Cauchyproblems have stimulated researchers to get either the new numerical methods or improve the existing ones [10-17]. Another reason is the computational effort and CPU time required bythe methods to solve the problem. The authors in [18]have tried toreduce the number of slope evaluations per integration step for autonomous initial value problems whereas the authors in [19] have extended the research work to non-autonomous type of problems. Moreover, nonlinear iterative integrators are suitable for the initial value problems having singular solutions along the integration interval under consideration as discussed in [20-24]. We consider the general first order ordinary differential equation with an initial condition, also called Cauchy Problem, as given below:
yx 0
dy fx ,y, 0 y 1
dx Existence of unique solution of 1 is assumed for the integration interval of x , . Here, exact
xx
0n
xn x0
solution is denoted by yx whereas the numerical solution is by y taking the step size h ,
nnN
where N 1,2,3,...
The generalform ofasingle-step explicit numericalintegrator tosolve an initial value problem is given as: y y hx ,y ;h2
n1 n fnn
Corresponding Author: Sania Qureshi, Department of Basic Sciences and Related Studies, Mehran University of EngineeringandTechnology, Jamshoro, Pakistan. Email:sania.qureshi@faculty.muet.edu.pk
where x ,y ;h can be expressed in terms of Taylor series expansion of an arbitraryfunction fx,y as
fnn
follows:
hp p
f xn,yn;h f fx,y3
p0 p1!x y Further, theTaylor series expansion of yxn h is ofthe form
11
2 3 22
yx h yx hf h f ff h f 2ff ff ff ff
n n xy xx xy yyyxy
2! 3! 4
f 3ff 3f 2 f 5ff f 3ff f3 f
14 xxx xxy xyy yxy xxy yyy 5
h Oh
2 32
4! 4f fyfyy 3ffxfyy ffy fxfy fxxfy
The proposed integrator ofthe present article is ofthe form: y y hx ,y ;h5
n1 n 3stageRK n n
where 3stageRK xn,yn;h b1k1 b2k2 b3k3 k1 fxn,yn , k2 fxn a2h,yn hk1b21 hc21 fy 2
k3 fxn a3h,yn hb31k1 b32k2 hc31k1 fy
Expanding k2and k3 in Taylor’s series, we obtain
1 22 212 2
k f ff b fa h f fb f fab ff c fa h
2 y 2,1 x 2 yy 2,1 x,y 2 2,1 y 2,1 x,x 2
,
22 133 122 21 2 13 34
fy y y f b f ,, f ab f , f f bc f ,, fab f ff ac f ,, ah Oh
, , 2,1 xyy 2 2,1 yy y 2,1 2,1 xxy 2 2,1 x,yy 2 2,1 xxx 2
62 26 1 22 21 22
f fb f fbb f fb f fab
yy 32 yy 31 32 yy 31 xy 3 32
22
f 32 ffy b f 3
k3 f fb y 31 xa h h2 21 22
f fab f fc fa f fbb ff ab
xy 3 31 y 31 xx 3 y 21 32 xy 2 32
2 1 321 32 1 221 221 21 2
f fbb f fbb f fab f fab f fab f fab
yyy 31 32 yyy 31 32 xyy 3 32 xyy 3 31 xx y 3 32 xxy 3 31
2 2 2222
122 3 1 2133 133
ff fbb f fb c f ff abb f f ab f fb f fb
y yy 21 32 y 32 21 y xy 2 21 32 y xx 2 32 yyy 32 yyy 31
34
2 266 h Oh
1
222 3
ffabb fff bc fff afffbb b fffabb fa
xyy 3 31 32 yy y 31 31 xy y 3 31 c y yy 21 31 32 x yy 2 31 32 xxx 3
6 222 2
ff fbb fff ab f ff abb ff aab f f f bc
y yy 21 32 x yy 2 32 y xy 3 21 32 x xy 2 332 yy y 32 31
3
Substituting theresult of k1,k and k into(5)then equatethe coefficientsofpowers ofhupto h with that
23 of 4 toobtain the followingorder conditions:
11
b b b 1 abb ab ab
123 2332 22 33
62 12211 1
ab ab bb bb bb abb abb abb 6
22 33 221 331 332 2221 3331 3332
262 3 1 222 1
1
bc bc bb b bb bb bb bb b
221 331 32132 221 331 332 33132
62 6
One of the solutions of the above nonlinear system 6 forms the proposed three-stage explicit RK iterative integrator ofthird order as given below:
k1 fx n,yn
22 2
k fx hy , hk hkf
2 nn 11 y
33 7
21 2
fx , hk hk 2
k hy hkf
3 nn 12 1 y
33 1
y y 2k k
hk
n1 n 1 23
4 The above proposed iterative integrator 7 can be used tosolve both autonomous and non-autonomous type of initial value problems in ordinarydifferential equations. After getting this new integrator, we will analyze it for its accuracy, convergence, order of consistency and linear stability. These are the important terms related to an iterative integrator for it to be acceptable in the field of computational and applied mathematics as proved in [25].
In order to obtain the local truncation error of the proposed integrator, a usual functional associated to the integratorhasbeen considered, that is given below:
Lzx,h zx h yn1
where zx is an arbitrary function defined along the integration interval x ,x and differentiable as many
0 n
times as required. Having expanded it into Taylor series about x and collecting the terms in h, the local truncation error under local assumption of the following form has been obtained that ensures at least third order accuracyofthe proposed integrator:
531121 1 3
ff ff ff ff ff
xy yyy ,, yx,xyxy x,,
24 72 24 72 216 4 5
T h Oh 8
n1
1 111 1
2
f ff f ,, f yfyf f , ff
x,, x,yx xxxx,fyy yy x
72 72 216 72 72
Definition4.1Given an initial value problem ;an iterative integrator with an
() fx ,yn ; yx y0 increment function x ,yh is saidtobe consistent if
nn; , fxy ,
lim ;
xyh
nn nn h0
The increment function ofthe proposed integrator 7 is shown as:
1
lim x ,yh lim k 2k k
;
nn 1 23
h0 h0
4
22 2
, 2fx hy hk hkf
fx y ,
nn nn 11 y
1 33
4 h0 21 2 x , hk hk 2h
lim
f hy kf
nn 12 1 y
33
f x n,yn
Thus, theproposed integrator isshown tobe consistentwith at least thirdorderaccuracy.
An iterative integrator shouldnot produce entirelydifferent results for verysmall changes in the input data, that is, it should be stable in order to be acceptable for use in solving practical problems in computational and applied mathematics. Numerical stabilityof an iterative integrator ensures the control of the magnitude of errors inherent to either the integrator or the initial value problem under consideration. Among various ways to check stabilityofthe iterative integrators, we consider Dahlquist’s test problem ofthe form
dy ; y y0,
yx 0 C
dx
Employing the proposed integrator 7 on this test problem, we obtain the following stability function whose
linear stabilityregion is shown bytheunshadedregion in theFigure1.
2 22 2 422 33
k y ; k y 1 h h ; k y 1 h h h
1 n2 n 3 n
3 33 Substituting all ofthese values in 7 ,the stabilityfunction is found tobe ofthe form:
121314
1 z z z z
Rz where z h .
264
In this section, some of the linear and nonlinear Cauchy problems in ordinary differential equations have been considered to show the behavior of the developed iterative integrator against other methods from well-established literature having same order of accuracy. Absolute maximum error, absolute error at the last nodal point of the given integration interval andCPU values for time have been presented to observe the performance of the developed method in comparison to other methods. Two standard methods called Runge-Kutta Method with Harmonic Mean of Three Quantities (RK3HM) [16] and Heun’s third order method [25] as shown below have been chosen tocomparethenumericalresultsobtained through thenewlydeveloped iterative integrator.
Heun’s Third Order RK3HMThirdOrder
0 000
0
00
0
13 13 0 0
23
23 00
0 23 0
23
23
23 43 0
14 0 34
kk k k kk
k2 k3
12
12 23
0
Table1.ErrorsandCPUtime values for CauchyProblem 1
Table2.ErrorsandCPUtime values for CauchyProblem 2
Table3.ErrorsandCPUvalues for CauchyProblem 3
Table4.ErrorsandCPUtime values for CauchyProblem 4
The newly developed third order iterative integrator is capable of solving Cauchy problems in the field of computational and applied mathematics. The maximum error and last error with step sizes 0.1, 0.05, 0.025 and 0.0125 are tabulated along-with the values of CPU timing in seconds. One may observe from these tabulated data that the absolute maximum and last error produced by the proposed iterative integrator are much smaller than the errors produced by other methods having same order of accuracy while consuming same amount of time on average. Similar sort of behavior has been observed while taking the step-size as large as 0.1 as shown by the Figures 2-6 for all the iterative integrators under consideration. The numerical results obtained through the proposed iterative integrator produce numerical values approximately close to the exact solution in comparison to the values obtained through Runge-Kutta Method with Harmonic Mean of Three Quantities and Heun’s third order method. For the proposed iterative integrator, small step size is also enough in comparison for other methods as shown in the Tables and the Figures above. Finally, it has been observed that the proposed iterative integrator is converging faster than the RK3HM and Heun’s third order method and it is the most effective integrator for solving the Cauchyproblems in ordinarydifferential equations as long as it is compared with theiterative integratorshavingsame order oflocal accuracyas that oftheproposed iterative integrator.
This paper develops anew single step Runge-Kutta iterative integrator for solvingCauchyproblems in ordinary differential equations. The integrator is found to be third order accurate and explicit in nature. Its linear stability analysis gives the stabilityregion which proves conditional stabilityof the proposed integrator. Examples in this paper proved that it is more accurate and effective integrator than some existing standardmethods. Tables 1to 5 above show the maximum error, the last error and CPU times related to all the integrators under consideration for the Cauchy problems with the variation in the step size. In addition, absolute errors produced by the above iterative integrators are smallest in case of the proposed integrator as shown by the Figures 2-6. The computations above evidently display the better accuracy of the integrator. The Runge-Kutta Method with Harmonic Mean grows faster in error than third order Heun and the proposed one. Hence, the proposed integrator performs best among the integrators taken for comparison. Based on the five Cauchy problems solved above, it follows that the proposed integrator is quite efficient specificallyin terms of local accuracy. It can be concluded that the proposed integrator is powerful and effective in finding numerical solutions Cauchy type problems arisingfrequentlyin the field ofcomputational and applied mathematics.
ACKNOWLEDGEMENT
The authors are highly thankful to the anonymous reviewers for their constructive criticism and suggestions to improve the first version ofthisresearch paper.
MATLABCODE
%ANewThird Order Iterative Integrator for CauchyProblems %NumericalProblem 1.y'=xy^3-y, y(0)=1; %Exact Solution: y(x)=2/sqrt(2+4*x+2*exp(2*x)); PartialDerivative f_y=3*x*y^2-1 %Integration Interval[0,1] clc; clear; close all;format shorte x(1)=0;y(1)=1;h=0.1; xfinal=1;N=ceil((xfinal-x(1))/h); Time_Proposed=cputime; fori=1:N
k1=f_der(x(i),y(i)); k2=f_der(x(i)+(2*h/3),y(i)+(2*h/3)*k1+(h^2)*k1*(3*x(i)*y(i)^2-1)); k3=f_der(x(i)+(2*h/3),y(i)-h*(k1/3-k2)-2*(h^2)*k1*(3*x(i)*y(i)^2-1));
y(i+1)=y(i)+(h/4)*(k1+2*k2+k3); x(i+1)=x(i)+h; end Time_Daud=cputime-Time_Proposed; t=x(1):h:xfinal; Exact=2./sqrt(2+4*t+2*exp(2*t)); Error_Daud=abs(Exact-y); Err_Max_Daud=max(Error_Daud); Err_Last_Daud=abs(Exact(length(t))-y(length(t))); semilogy(t,Error_Daud,'ko-'),hold on %%
%ThirdOrder Heun Method TIME=cputime; fori=1:N
k1=f_der(x(i),y(i)); k2=f_der(x(i)+h*(1/3),y(i)+(1/3)*h*k1); k3=f_der(x(i)+(2/3)*h,y(i)+(2/3)*h*k2);
y(i+1)=y(i)+h*(1/4)*(k1+3*k3); x(i+1)=x(i)+h; end Time_Heun3=cputime-TIME; Error_Heun3=abs(Exact-y); Err_Max_Heun3=max(Error_Heun3); Err_Last_Heun3=abs(Exact(length(x))-y(length(x))); semilogy(t,Error_Heun3,'r*-') %% %3rd order MODIFIEDRKRule usingHarmonic Mean RK3HM=cputime; fori=1:N
k1=f_der(x(i),y(i)); k2=f_der(x(i)+(2/3)*h,y(i)+(2/3)*h*k1); k3=f_der(x(i)+(2/3)*h,y(i)-h*(2/3)*k1+h*(4/3)*k2); y(i+1)=y(i)+h*((k1*k2)/(k1+k2)+(k2*k3)/(k2+k3));
x(i+1)=x(i)+h; end Time_RK3HM=cputime-RK3HM; Error_RK3HM=abs(Exact-y); Err_Max_RK3HM=max(Error_RK3HM); Err_Last_RK3HM=abs(Exact(length(x))-y(length(x))); semilogy(t,Error_RK3HM,'b>-') %% functiondydx=f_der(x,y) dydx=x.*y.^3-y; end
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