International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 2, April 2017, Pages: 58-63

Development of a New One-Step Scheme for the Solution of Initial Value Problem (IVP) in Ordinary Differential Equations

Fadugba Sunday Emmanuel1, *, Falodun Bidemi Olumide2

1Department of Mathematics, Ekiti State University, Ado Ekiti, Nigeria

2Department of Mathematics, University of Ilorin, Ilorin, Nigeria

Email address:

(F. S. Emmanuel)

*Corresponding author

To cite this article:

Fadugba Sunday Emmanuel, Falodun Bidemi Olumide. Development of a New One-Step Scheme for the Solution of Initial Value Problem (IVP) in Ordinary Differential Equations. International Journal of Theoretical and Applied Mathematics. Vol. 3, No. 2, 2017, pp. 58-63.
doi: 10.11648/j.ijtam.20170302.12

Received: October 27, 2016; Accepted: January 16, 2017; Published: February 9, 2017


Abstract: In this paper, a new one-step scheme was developed for the solution of initial value problems of first order in ordinary differential equations. In its development a combination of interpolating function and Taylor series were used. The method was used for the solution of initial value problems emanated from real life situations. The numerical results showed that the new scheme is consistent, robust and efficient.

Keywords: Interpolating Function, Initial Value Problem, One-Step Method, Ordinary Differential Equation, Taylor Series


1. Introduction

In the past years, a large number of methods suitable for solving ordinary differential equations have been proposed. A major impetus to developing numerical procedures was the invention of calculus by Newton and Leibnitz, as this led to accurate mathematical models for physical reality, such as Sciences, Engineering, Medicine and Business. These mathematical models cannot be usually solved explicitly and numerical method to obtain approximate solutions is needed.

The approach for the solution of initial value problems in ordinary differential equations based on numerical approximations were developed before the existence of programmable computers.

A numerical method is a complete and unambiguous set of procedures for the solution of a problem, together with computable error estimates. The study and implementation of such methods is the province of numerical analysis [12].

Development of numerical integrator for the solution of initial value problems in ordinary differential equations has attracted the attention of many researchers in recent years. There are numerous methods that produce numerical approximations to solution of initial value problem in ordinary differential equation such as Euler’s method which was the oldest and simplest method originated by Leonhard Euler in 1768, Improved Euler method, Runge Kutta methods described by Carl Runge and Martin Kutta in 1895 and 1905 respectively.

There are many excellent literature and exhaustive texts on this subject that may be consulted, such as [1], [3], [4], [5], [6], [7], [8], [9], [11], [12] just to mention a few.

In this paper we consider the initial value problem of the form

(1)

and develop an algorithm which can effectively solve initial value problems in ordinary differential equation. Many researchers have solved the problem in (1). However, if the solution to (1) posseses a singularity point, a numerical integration formulae will be more effective. In another development, [10] recently discussed one-step method of Euler-Maruyama type for the solution of stochastic differential equations using varying step sizes. [2] derived a continuous linear multistep method using Hermite polynomials as basis functions.

In this paper we develop a new accurate scheme for the solution of the initial value problems in ordinary equation. The rest of the paper is organized as follows: Section Two is the development of a new scheme. Section Three consists of some basic concepts vital to the development of the new scheme. Section Four consists of numerical experiment and discussion of results. Section Five concludes the paper.

2. Development of the New Scheme

We develop a new one-step scheme in solving an initial value problem in ordinary differential equations as follows. We intend to solve problem (1) with power series polynomial such that

(2)

With integration interval of [a,b] in the form  with step size  given by  such that  

Expanding (2) above we obtain the interpolant of the form:

(3)

where  and  are real undetermined coefficients.

At  becomes

(4)

Also,

(5)

Differentiating (4) and let  yields

(6)

Similarly,

(7)

Simplifying (6) and (7) above we obtain

(8)

and

(9)

Since

(10)

and

(11)

It worth mentioning in the present paper that  varies which makes our derived scheme an avenue to solve any problem whose initial condition is not limited only to .

Subtracting (4) from (5) and using (8) and (9) with  yields

(12)

Using the fact that

(13)

Substituting (12) into (13) we obtain

(14)

Equation (14) is the new one-step scheme.

3. Some Basic Concepts [9]

We consider the following basic concepts which are very vital to the development of the new scheme.

3.1. Stability

A numerical method is said to be stable if the difference between the numerical solution and the exact solution can be made as small as possible, that is if there exists two positive numbers  and  such that the following holds.

(15)

3.2. Consistency

A numerical method with an increment function  is said to be consistent with the initial value problem (1) under consideration, if

(16)

3.3. Convergence

A numerical method is said to be convergent if for all initial value problem satisfying the hypothesis of the Lipschitz condition given by

(17)

where the Lipschitz constant is denoted by . The necessary and sufficient conditions for convergence are the stability and consistency.

3.4. Round off Error

This can be defined as the error due to computing device. They arise because it is possible to represent all real numbers exactly on a finite-state machine. It can be represented mathematically as

(18)

where  is the approximate solution and  is the computer output. The magnitude depends in the storage and the arithmetic operation adopted. In such cases, double precision are employed to guarantee an adequate approximation.

4. Numerical Experiment and Discussion of Results

This section presents a comparative study of the new scheme and the theoretical solution with different step sizes.

4.1. Numerical Experiment

It is usually necessary to demonstrate the suitability and applicability of the newly developed scheme. In this course, we translated the algorithm of our scheme into MATLAB programming language and implemented with a problem emanated from real life situations. The performance of the method was checked by comparing its accuracy and efficiency. The efficiency was determined from the number of iterations counts and number of functions evaluations per step while the accuracy is determined by the size of the discretization error estimated from the difference between the exact solution and the numerical approximations.

4.1.1. Application of the New Scheme to Real Life Problem

Let us assume that a colony of 1000 bacteria are multiplying at the rate of  per hour per individual (i.e an individual produces an average of  off spring every hour, see (Hahn B. D (1997)). How many bacteria are there after  hour? If we assume that the colony grows continuously and without restriction.

Solution

It is possible to model this growth with a differential equation of first order of the form:

(19)

with initial condition:

(20)

Equations (19) and (20) connote the IVP given by

(21)

where  is the population size at time . The exact solution to (21) is obtained as

(22)

Take  and  The results generated from our scheme for h = 0.0125 and h = 0.01 are shown in Tables 1 and 2 below respectively.

4.1.2. Table of Results

Table 1. Comparative Analyses of the Results of the New Scheme with Theoretical Solution for h = 0.0125.

t-value Exact-solution Computed-solution Error
0.0125000 1010.050167084167900000 1010.049792316084100000 3.747681e-004
0.0250000 1020.201340026755800000 1020.200582957764600000 7.570690e-004
0.0375000 1030.454533953516800000 1030.453386937238100000 1.147016e-003
0.0500000 1040.810774192388200000 1040.809229467362700000 1.544725e-003
0.0625000 1051.271096376024200000 1051.269146064173400000 1.950312e-003
0.0750000 1061.836546545359600000 1061.834182650425400000 2.363895e-003
0.0875000 1072.508181254216700000 1072.505395660181200000 2.785594e-003
0.1000000 1083.287067674958700000 1083.283852144445700000 3.215531e-003
0.1125000 1094.174283705210400000 1094.170629877864900000 3.653827e-003
0.1250000 1105.170918075647700000 1105.166817466496500000 4.100609e-003
0.1375000 1116.278070458871300000 1116.273514456662300000 4.556002e-003
0.1500000 1127.496851579375700000 1127.491831444896900000 5.020134e-003
0.1625000 1138.828383324621900000 1138.822890188999500000 5.493136e-003
0.1750000 1150.273798857227300000 1150.267823720201700000 5.975137e-003
0.1875000 1161.834242728283000000 1161.827776456463700000 6.466272e-003
0.2000000 1173.510870991810200000 1173.503904316909000000 6.966675e-003
0.2125000 1185.304851320365700000 1185.297374837407900000 7.476483e-003
0.2250000 1197.217363121810200000 1197.209367287323600000 7.995834e-003
0.2375000 1209.249597657251600000 1209.241072787431700000 8.524870e-003
0.2500000 1221.402758160169900000 1221.393694429024000000 9.063731e-003
0.2625000 1233.678059956743500000 1233.668447394210500000 9.612563e-003
0.2750000 1246.076730587381000000 1246.066559077428300000 1.017151e-002
0.2875000 1258.600009929478100000 1258.589269208173800000 1.074072e-002
0.3000000 1271.249150321404800000 1271.237829974968000000 1.132035e-002
0.3125000 1284.025416687741700000 1284.013506150565900000 1.191054e-002
0.3250000 1296.930086665772000000 1296.917575218426000000 1.251145e-002
0.3375000 1309.964450733247500000 1309.951327500450600000 1.312323e-002
0.3500000 1323.129812337437000000 1323.116066286008800000 1.374605e-002
0.3625000 1336.427488025472300000 1336.413107962257300000 1.438006e-002
0.3750000 1349.858807576003100000 1349.843782145770500000 1.502543e-002
0.3875000 1363.425114132178100000 1363.409431815493000000 1.568232e-002
0.4000000 1377.127764335957400000 1377.111413447029000000 1.635089e-002
0.4125000 1390.968128463780400000 1390.951097148280700000 1.703132e-002
0.4250000 1404.947590563594100000 1404.929866796450400000 1.772377e-002
0.4375000 1419.067548593257500000 1419.049120176418500000 1.842842e-002
0.4500000 1433.329414560340600000 1433.310269120513300000 1.914544e-002
0.4625000 1447.734614663324700000 1447.714739649685000000 1.987501e-002
0.4750000 1462.284589434224900000 1462.263972116098100000 2.061732e-002
0.4875000 1476.980793882643000000 1476.959421347157100000 2.137254e-002
0.5000000 1491.824697641270600000 1491.802556790979700000 2.214085e-002
0.5125000 1506.817785112853700000 1506.794862663332400000 2.292245e-002
0.5250000 1521.961555618633800000 1521.937838096041300000 2.371752e-002
0.5375000 1537.257523548281600000 1537.232997286896800000 2.452626e-002
0.5500000 1552.707218511335900000 1552.681869651061600000 2.534886e-002
0.5625000 1568.312185490168800000 1568.285999974004000000 2.618552e-002
0.5750000 1584.073984994481600000 1584.046948565965100000 2.703643e-002
0.5875000 1599.994193217360400000 1599.966291417979800000 2.790180e-002
0.6000000 1616.074402192893300000 1616.045620359466100000 2.878183e-002
0.6125000 1632.316219955378800000 1632.286543217396200000 2.967674e-002
0.6250000 1648.721270700127700000 1648.690683977070000000 3.058672e-002
0.6375000 1665.291194945886000000 1665.259682944502200000 3.151200e-002
0.6500000 1682.027649698885900000 1681.995196910442500000 3.245279e-002
0.6625000 1698.932308618550200000 1698.898899316043500000 3.340930e-002
0.6750000 1716.006862184857900000 1715.972480420193700000 3.438176e-002
0.6875000 1733.253017867394600000 1733.217647468532300000 3.537040e-002
0.7000000 1750.672500296100200000 1750.636124864163100000 3.637543e-002
0.7125000 1768.267051433734400000 1768.229654340082300000 3.739709e-002
0.7250000 1786.038430750072600000 1785.999995133341400000 3.843562e-002
0.7375000 1803.988415397856000000 1803.948924160958800000 3.949124e-002
0.7500000 1822.118800390508100000 1822.078236197599800000 4.056419e-002
0.7625000 1840.431398781636300000 1840.389744055042700000 4.165473e-002
0.7750000 1858.928041846340900000 1858.885278763447100000 4.276308e-002
0.7875000 1877.610579264342100000 1877.566689754445900000 4.388951e-002
0.8000000 1896.480879304950100000 1896.435845046075400000 4.503426e-002
0.8125000 1915.540829013894800000 1915.494631429566000000 4.619758e-002
0.8250000 1934.792334402030100000 1934.744954658007100000 4.737974e-002
0.8375000 1954.237320635938000000 1954.188739636911600000 4.858100e-002
0.8500000 1973.877732230446100000 1973.827930616692600000 4.980161e-002
0.8625000 1993.715533243080700000 1993.664491387076400000 5.104186e-002
0.8750000 2013.752707470474900000 2013.700405473468000000 5.230200e-002
0.8875000 2033.991258646748700000 2033.937676335290700000 5.358231e-002
0.9000000 2054.433210643886000000 2054.378327566319100000 5.488308e-002
0.9125000 2075.080607674120500000 2075.024403097024800000 5.620458e-002
0.9250000 2095.935514494362500000 2095.877967398956600000 5.754710e-002
0.9375000 2117.000016612672400000 2116.941105691172700000 5.891092e-002
0.9500000 2138.276220496816200000 2138.215924148750700000 6.029635e-002
0.9625000 2159.766253784912400000 2159.704550113389400000 6.170367e-002
0.9750000 2181.472265498198800000 2181.409132306130700000 6.313319e-002
0.9875000 2203.396426255934300000 2203.331841042216600000 6.458521e-002
1.0000000 2225.540928492464900000 2225.474868448106000000 6.606004e-002

Table 2. Comparative Analyses of the Results of the New Scheme and Theoretical Solution for h = 0.01.

t-value Exact-solution Computed-solution Error
0.0100000 1008.032085504273500000 1008.031893599467600000 1.919048e-04
0.0200000 1016.128685406094900000 1016.128298513728300000 3.868924e-04
0.0300000 1024.290317890621500000 1024.289732890798600000 5.849998e-04
0.0400000 1032.517505305118200000 1032.516719040404500000 7.862647e-04
0.0500000 1040.810774192388200000 1040.809783467408600000 9.907250e-04
0.0600000 1049.170655324470500000 1049.169456905503700000 1.198419e-03
0.0700000 1057.597683736611300000 1057.596274351180000000 1.409385e-03
0.0800000 1066.092398761505100000 1066.090775097962000000 1.623664e-03
0.0900000 1074.655344063813600000 1074.653502770922800000 1.841293e-03
0.1000000 1083.287067674958700000 1083.285005361474100000 2.062313e-03
0.1100000 1091.988122028197500000 1091.985835262436000000 2.286766e-03
0.1200000 1100.759063993978800000 1100.756549303389600000 2.514691e-03
0.1300000 1109.600454915582500000 1109.597708786311600000 2.746129e-03
0.1400000 1118.512860645045300000 1118.509879521496300000 2.981124e-03
0.1500000 1127.496851579375700000 1127.493631863766300000 3.219716e-03
0.1600000 1136.553002697060300000 1136.549540748973200000 3.461948e-03
0.1700000 1145.681893594861800000 1145.678185730792600000 3.707864e-03
0.1800000 1154.884108524913700000 1154.880151017813400000 3.957507e-03
0.1900000 1164.160236432112500000 1164.156025510925500000 4.210921e-03
0.2000000 1173.510870991810200000 1173.506402841008400000 4.468151e-03
0.2100000 1182.936610647811000000 1182.931881406921300000 4.729241e-03
0.2200000 1192.438058650669500000 1192.433064413799700000 4.994237e-03
0.2300000 1202.015823096301600000 1202.010559911658400000 5.263185e-03
0.2400000 1211.670516964900800000 1211.664980834305400000 5.536131e-03
0.2500000 1221.402758160169900000 1221.396945038567500000 5.813122e-03
0.2600000 1231.213169548867700000 1231.207075343832200000 6.094205e-03
0.2700000 1241.102379000671800000 1241.095999571905600000 6.379429e-03
0.2800000 1251.071019428362400000 1251.064350587192200000 6.668841e-03
0.2900000 1261.119728828329500000 1261.112766337195600000 6.962491e-03
0.3000000 1271.249150321404800000 1271.241889893346100000 7.260428e-03
0.3100000 1281.459932194021300000 1281.452369492155400000 7.562702e-03
0.3200000 1291.752727939704100000 1291.744858576702000000 7.869363e-03
0.3300000 1302.128196300894400000 1302.120015838449300000 8.180462e-03
0.3400000 1312.587001311108500000 1312.578505259400800000 8.496052e-03
0.3500000 1323.129812337437000000 1323.120996154592600000 8.816183e-03
0.3600000 1333.757304123384500000 1333.748163214927900000 9.140908e-03
0.3700000 1344.470156832052900000 1344.460686550355400000 9.470282e-03
0.3800000 1355.269056089671900000 1355.259251733394900000 9.804356e-03
0.3900000 1366.154693029480100000 1366.144549843011600000 1.014319e-02
0.4000000 1377.127764335957400000 1377.117277508843200000 1.048683e-02
0.4100000 1388.188972289412500000 1388.178136955782700000 1.083533e-02
0.4200000 1399.339024810930800000 1399.327836048918700000 1.118876e-02
0.4300000 1410.578635507678700000 1410.567088338836800000 1.154717e-02
0.4400000 1421.908523718577500000 1421.896613107285200000 1.191061e-02
0.4500000 1433.329414560340600000 1433.317135413206400000 1.227915e-02
0.4600000 1444.842038973879400000 1444.829386139138900000 1.265283e-02
0.4700000 1456.447133771086600000 1456.434102037992500000 1.303173e-02
0.4800000 1468.145441681989700000 1468.132025780197900000 1.341590e-02
0.4900000 1479.937711402288600000 1479.923906001235100000 1.380540e-02
0.5000000 1491.824697641270600000 1491.810497349545400000 1.420029e-02
0.5100000 1503.807161170112100000 1503.792560534825700000 1.460064e-02
0.5200000 1515.885868870569100000 1515.870862376712500000 1.500649e-02
0.5300000 1528.061593784057300000 1528.046175853855400000 1.541793e-02
0.5400000 1540.335115161127300000 1540.319280153386900000 1.583501e-02
0.5500000 1552.707218511336400000 1552.690960720787400000 1.625779e-02
0.5600000 1565.178695653522000000 1565.162009310151900000 1.668634e-02
0.5700000 1577.750344766478300000 1577.733224034859900000 1.712073e-02
0.5800000 1590.422970440039800000 1590.405409418652900000 1.756102e-02
0.5900000 1603.197383726574600000 1603.179376447121100000 1.800728e-02
0.6000000 1616.074402192893800000 1616.055942619605000000 1.845957e-02
0.6100000 1629.054849972574900000 1629.035932001512900000 1.891797e-02
0.6200000 1642.139557818705700000 1642.120175277058500000 1.938254e-02
0.6300000 1655.329363157055700000 1655.309509802422800000 1.985335e-02
0.6400000 1668.625110139667400000 1668.604779659342700000 2.033048e-02
0.6500000 1682.027649698886800000 1682.006835709129700000 2.081399e-02
0.6600000 1695.537839601820500000 1695.516535647122700000 2.130395e-02
0.6700000 1709.156544505233600000 1709.134744057578500000 2.180045e-02
0.6800000 1722.884636010888000000 1722.862332469002000000 2.230354e-02
0.6900000 1736.722992721326400000 1736.700179409923500000 2.281331e-02
0.7000000 1750.672500296101600000 1750.649170465120300000 2.332983e-02
0.7100000 1764.734051508460200000 1764.710198332292300000 2.385318e-02
0.7200000 1778.908546302479000000 1778.884162879192700000 2.438342e-02
0.7300000 1793.196891850663400000 1793.171971201216400000 2.492065e-02
0.7400000 1807.600002612005300000 1807.574537679452100000 2.546493e-02
0.7500000 1822.118800390509700000 1822.092784039200100000 2.601635e-02
0.7600000 1836.754214394190500000 1836.727639408960600000 2.657499e-02
0.7700000 1851.507181294539100000 1851.480040379894700000 2.714091e-02
0.7800000 1866.378645286473300000 1866.350931065763900000 2.771422e-02
0.7900000 1881.369558148764100000 1881.341263163351400000 2.829499e-02
0.8000000 1896.480879304952200000 1896.451996013367500000 2.888329e-02
0.8100000 1911.713575884749200000 1911.684096661844700000 2.947922e-02
0.8200000 1927.068622785936000000 1927.038539922027000000 3.008286e-02
0.8300000 1942.547002736755100000 1942.516308436754100000 3.069430e-02
0.8400000 1958.149706358806700000 1958.118392741348800000 3.131362e-02
0.8500000 1973.877732230448600000 1973.845791327007600000 3.194090e-02
0.8600000 1989.732086950705000000 1989.699510704703000000 3.257625e-02
0.8700000 2005.713785203689200000 2005.680565469595900000 3.321973e-02
0.8800000 2021.823849823545300000 2021.789978365967500000 3.387146e-02
0.8900000 2038.063311859907300000 2038.028780352672800000 3.453151e-02
0.9000000 2054.433210643888700000 2054.398010669118300000 3.519997e-02
0.9100000 2070.934593854599400000 2070.898716901770500000 3.587695e-02
0.9200000 2087.568517586197700000 2087.531955051199500000 3.656253e-02
0.9300000 2104.336046415478900000 2104.298789599659400000 3.725682e-02
0.9400000 2121.238253470012800000 2121.200293579212500000 3.795989e-02
0.9500000 2138.276220496819900000 2138.237548640399800000 3.867186e-02
0.9600000 2155.451037931604800000 2155.411645121465900000 3.939281e-02
0.9700000 2172.763804968546500000 2172.723682118135000000 4.012285e-02
0.9800000 2190.215629630644300000 2190.174767553951400000 4.086208e-02
0.9900000 2207.807628840633700000 2207.766018251183600000 4.161059e-02
1.0000000 2225.540928492468500000 2225.498560002297400000 4.236849e-02

4.2. Discussion of Results

The numerical experiment shows that the new scheme provides comparable results. The differences between our scheme and the theoretical solution are negligible from a practical point of view as we can see from Tables 1 and 2 above. Also we deduce that the smaller the step size, the more accurate is the new scheme. The numbers of bacteria present after 1 one hour for h = 0.0125 and h = 0.01 were obtained as 2225.474868448106000000 and 2225.498560002297400000 respectively.

5. Conclusion

In this paper, we have developed a new scheme for the solution of initial value problems in ordinary differential equations. The new scheme was used to obtain numerical solution to real life problem. The comparative analyses of the results were also presented. The numerical results of our scheme were compared favorably with the theoretical solution. Hence the new scheme is computational efficient, robust and easy to implement.


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