Solution of 2nd Order Ode by Numerical Methods Such as Picard’s and Runge-Kutta

Introduction

The word differential and equation itself defines as an equation of derivatives or differentials. Differential equation (or “DE”) is the most important branch of mathematics in which mathematical equation is exhibited in form of functions and its derivatives of one or more dependent variables. Differential equation reveals correlation that involves the rate of change. such connection builds some precise for advanced ideas which led cautiously to mathematical model involving function and their respective derivatives .Differential equation is immensely interesting branch and it plays major role in field of engineering, physics[1,2] as well as biology[3,4], economics[5], finance and various aspect of human knowledge because we can reduced the real life problem into differential equations satisfying given conditions.

In seventeenth century some mathematician like Rene Discartes(1596-1650), Isaac Newton(1642-1727) and Leonhard Euler(1707-1783) proposed Modern applied mathematics. Before this century people were unfamiliar with the name of numerical mathematics. Sir Isaac Newton and Euler introduced this term. November 1947 is known as the birthday of modern numerical analysis.

Sir Isaac Newton was the first mathematician who solved the differential equation in the seventeenth century (1642-1727), he introduced the equations as fluxional equation and according to some mathematician, the term differential equation has introduced in 1675 by Gottfried Wilhelm vonLeibnitz who was a fellow of Newton, by writing the mathematical equation[6]

dx = (1/2)

In this century most of the mathematician used different techniques for obtaining the result of differential equations but Augustin-Louis Cauchy was the only one who developed a general theory for differential equations.

After that, the search for normal technique of integrating differential equation started during the classification of first order differential equation by Sir Isaac Newton. He classified the equation into 3 classes[7].

=f(x)

- (1.1)

+ y = u

In 1690, James Bernoulli had solved a problem of Isochrone in which he had described how to find a curve along which a body will fall with uniform vertical velocity. In 1694, G.W Leibniz had addressed a problem of quadrature of the hyperbola for finding a area under the curve on a given interval. In 1696, John Bernoulli had solved Brachistochrone problem in which he addressed how to find the equation of the path down which a particle will fall from one point to another in the shortest time. In 1698 John Bernoulli had solved a problem of orthogonal trajectories in which he described how to find a curve that cuts all the curves of a family of curve at right angles. In 1728, Leonhard Euler had reduced the 2nd order equations into 1st order, In 1734, Alexis Clairaut had solved a problem of singular solution in which he described how to find an equation of an envelope of the family of curves represented by the general solution. In 1743, Joseph Lagrange had addressed the problem of determining integrating factor for the general linear equation. In 1762, Jean d’Alembert had addressed the problem of linear equation with constant coefficient in which he described the conditions under which the order of linear differential equation could be covered. [8]

Differential equation is categorized into 2 parts.

Ordinary differential equation In which equation is expressed in one or more dependent variable at least one of its derivatives with respect to one individualistic variable and whereas the limitation of partial differential equation is just containing two or more than two independent variables.

Here the first two equations show themselves as ordinary differential equation whereas other shows itself as partial differential equation. Differential equations contain equations of first, second and higher order (nth) with one, two and n arbitrary constants in its general solutions respectively.

In Pure Mathematics we have studied differential equation with different perspective that is mostly concerned with their solutions. Mostly simplest equations can be solved by explicit formulae. Hence for solving complicated equation, numerical methods are developed.

In this paper it is intended to discuss the numerical solution of ordinary differential equation of second order by numerical methods such as Picard’s method and Runge- Kutta method.

History of Picard’s method

French mathematician Charles Emile Picard’s (1856-1941) developed this method so on the name of mathematician this iterative method is known as Picard’s method. Picard iteration is a successive approximation that is used for establishing the existence of solution of differential equation that passes through some point. This iterative method helps in construction of appropriate solution of a differential equation y’ = f(x,y(x)). It is the oldest method to know the existence of solution of any ordinary differential equations

Let us examine the first order ordinary differential equation

= f(x, y(x))

With an initial condition y()= -(1.2)

Here f(x,y) is the function of variable x and y.

Integrating equation 1.2 between respective limits for x and y

We get,

=

y -=

y = + -(1.3)

Here we can see that, in equation 1.3 ,the integrand is an unknown function of y. Which is known as integral equation and such equation can be solved by iteration method or process of successive approximations.

Now replace y by on R.H.S in equation 1.3

We have,

= +

It is the first approximation of y. Here the function under the integral sign of x only that’s why integration is possible and we get first estimation of y.

For improvement in the solution we can again integrate the above equation and replace by and get second estimations of y.

We get,

= +

Proceeding in similar manner

we get sequence of estimation

, ………….

The nth approximation is given by

= +

This process is replicated as many times until we get the same value of and or we can leave when integration becomes complicated. After each iterations we obtain a better approximation of desired solution than the previous one.

Disadvantage: The biggest disadvantage of this method is unsatisfactory solution. Sometimes it is so difficult to analyse the complicated integration and we can carry only few successive integration exactly so we can’t obtain the accurate solution.

Further, mathematician Chebyshev refined this Picard’s method and introduced new method that is known as The Modified ChebyshevPicard Iteration (MCPI). It is an iterative path approximation method which helps to find the solution of nonlinear ordinary differential equations.

History of Runge-Kutta method

Many methods developed to solve ordinary differential equation but one of the standard and important numerical techniques to solve ODE is Runge-kutta method. Two German mathematician C.D.T Runge (1856-1927) and M.W. Kutta(1867- 1944) attributed Runge-Kutta method in 1901. It is the refinement of improved Euler method. During the study of atomic spectra, Carl Runge introduced the numerical method for solving the differential equation and now days these numerical methods are still applied. He was a mathematician and physicist too as he did so much work on several research paper in the field of mathematics as well as physics. Kutta was another applied mathematician who contributed in differential equations in the theory of airfoil lift in aerodynamics which based on Kutta- Joukowski. So their name is associated with this Runge-Kutta method.

It is applicable for higher order differential equation as well as first order and second order. The result of this method gives satisfaction for accurate solution without the need of the values of step size h. Runge-Kutta developed sequence of formulae which is known as second order approximation , third order approximation and fourth order approximation for solving the differential equation of any order. In 1895 Carl Runge published the first Runge-Kutta method and Martin Kutta developed the suitable and popular fourth order Runge-Kutta method in 1901.These two German mathematician refined the modified Euler method by replacing the average slope of two points with a weighted average slope of function f(x,y(x)) at 4 points in the interval and this refinement leads gradually to the order of accuracy from to . [9] To solve differential equation or coupled of differential equations, this method can be applied by using sequence of numerical formulae.

Here three approximations are presented of order two, three and four.

Runge-kutta method of order 2

This method is slightly similar to Euler method so it can be considered as Euler method. [ numerical solution of de NOTE] {ref }

Let us consider the initial value problem of differential equation of first order

= f(x,y(x)) , y() =

Suppose h be the step size between the equidistant values of x.

The first increment in y is computed by the formula

= hf()

= hf(

y = () …………..(1.4)

Here y is 1st increment of y. we can easily find the increment in y by using this quadratic formula.

This formula is known as second order Runge-Kutta formula or quadratic formula of Runge-kutta method.

This method is also applicable for finding the value of y at any point by using formula

= +h and =y where y is defined in 1.4 equation.

Similarly, we can compute the another increment in y for second interval only by replacing ,) with , )

= hf()

= hf()

= ()

In same manner we can find all increment in y for different intervals by using the initial value of that particular interval.

This 2nd order approximation method does not give an accurate solution and it is only applicable for first order differential equation so another formula has been developed by Runge-Kutta which is known as Third order Runge-Kutta formula that is presented with list of several formulae.

= f()

= hf()

= f(f()

= ()

Taken as same order as given

Then = +h and = + y

We can also generalise this formula for successive approximations in different interval.

This formula gives better approximation than preceding formula. But this method also does not give accurate solution.

Furthermore, Runge and Kutta developed a suitable formula which gives sufficient accuracy without taking extremely small values of step size.

Runge-Kutta 4th order approximation formula

It is the most suitable and widely applicable formula that can be used in any case even when differential equation is very complicated to solve and whenever we need to obtain highly accurate solution. It is generally superior to second order Runge-Kutta formula. [10]

We consider the initial value problem of differential equation of first order.

= f(x,y(x)) , y() =

And h is the interval between equidistant values of x as defined in above two formulae.

Then the list of several formulae are given below for the first increment in y

= hf()

= hf()

= hf()

=h f()

= ()

Taken in same order

Then, = +h and = + y

It can be further generalised for another successive approximation in y for different intervals by using the replacement of x and y for the different intervals in same formulae.

Hence, to obtain increment in y for the nth interval we have to just put the value of and in the above mentioned formulae for , , etc…

For solving the initial value problem of ordinary differential equation and obtaining the value of y at any given point we use general formula which is given below

= + ( , ,h)

Where = + h

And = ()

= hf()

= hf()

= hf()

=h f()

Where h and are the step size.

Need of Numerical Methods:

There exist many analytical methods for solving ordinary differential equation in differential equation. Such as separation of variables method, Exact method, homogenous method, using an integrating factor method etc.

In 1993, method for finding the solution of boundary value problem for second order differential equations was introduced by mathematician Zhang [11]. Yang introduced a method for finding the solution of 2nd order differential equation having periodic coefficients in 2003 [12]. In 2005, mathematicians Nieto and Lopez presented green’s function for solving higher order differential equation containing periodic value problem [13]. Further, green function method was also applied by other mathematicians Yang etal [14]. In 2008, periodic solution for higher order differential equation having deviated argument was obtained by Pan [15]. In 2009, Lopez solved 2nd order functional differential equation by using non-local boundary value problem [16].

Further, For a given general equation of 1st order and 1st degree = f(x,y), many analytical methods are applicable. But these symbolic methods are applicable for solving only some category of differential equations. Sometimes these methods can’t be applied for some real life problem that express into mathematical modulation. Hence for solving and obtaining the solution of such type of Ode’s, we need to apply numerical methods. Such as Picard’s, Runge-Kutta, Euler method, Modified Euler method etc. Numerical techniques are more popular and much efficient for finding the numerical accurate values of dependent and independent variables of ordinary differential equations not for the relation of x and y. The concept of these numerical techniques is much similar to the numerical approaches of integration like Simpson and Trapezoidal method.

By using numerical methods, we can solve initial value problem as well as boundary value problem.

Initial value problem involves the problem in which the function and its respective derivative is defined at a initial point only however boundary value problem contains the problem in which function and their respective higher order derivatives are defined at two or more points.

Review of Literature

Famelis, Papakostas & Tsitouras(2018) have compared the result obtained by Sofroniou code and New symbolic code. They have applied Mathematica 4.0 software to build in functions Timing and MemoryInUse. They have concluded the result that new code requires less memory as comparison to Sofroniou code and it is more efficient method. For high order methods, Sofroniou code takes much time and tough to use. Whereas new code time measurements endure small and much easy to use and understand. Furthermore, Sofroniou code has not worked for order 17 methods.

T.S Mohamed et. al (2018) have proposed a special class of explicit Runge-Kutta –Nystrom (RKN) methods for solving second order ordinary differential equations along with 3rd derivatives and symbolized as STDRKN. The methods include single estimation of 2nd order and much evaluations of 3rd order derivative in each step. The stability region of methods is also considered through numerical experiments. New special explicit two- derivative Runge-Kutta-Nystrom method of order 5 (STDRKN5(3)) and new special explicit two-derivative RKN method of order 4 (STDRKN4(2)) are derived by T.S Mohamed etal . In this paper, Special class of explicit two derivative RKN methods till 5th order that contain one f-evaluation and lowest number of g evaluation was formulated. They explained how much the STDRKN methods are convenient than the general classical Runge-Kutta-Nystrom methods and TDRKN methods. They concluded that, this new method is applicable for higher order with minor functions evaluations of each stage and also give higher accuracy at every stage than RKN.

C. Senthilnathan(2018) has published a research paper in which he focused two applicable methods for solving initial value problem of ordinary differential equation : Euler method and Runge-Kutta 4th order method. Also he has compared the solution which obtained by these two presented methods with the exact solution and performance and numerical collation among Runge –Kutta 2nd order, 3rd order, 4th order and Euler method through MATLAB software are also presented. He has concluded 4th order Runge-Kutta method is more realistic than Euler method and the solution of 4th order Runge-Kutta method is much closer to exact solution and this method gives minor error.

The research article by Claudio fariaet. al(2018) is focused on implementation of numerical methods by using MATLAB software for finding the solution of differential equations ,the equation which is impossible or take much time to solve by using analytical method. In his paper he addressed that MATLAB software is extremely useful tool not only for finding the solution of problem but for the graphical representation of problems by using various numerical method. By graphical representation. We can easily understand the difference of solution which obtained by different numerical method such as Euler method, Runge-kutta method etc.

Abbas Fadhil Abbas A1-Shimmary (2017) has developed some unpretentious method for the trees representation that can be applied in the formulation of Runge-Kutta method and also presented a derivation of Runge-Kutta method of six order in 7stages. He has used figurative computational study to clarify the method.

Adam Bashforthmulti-step methods have been introduced to find the approximate solution of higher order differential equations. ShabanGholamtabar and NouredinParandin(2014) used order reduction method for converting the equation into 1st order differential equation and then used single step method for appropriate initial order. In the respective research article Gholamtabar and Nouredin compared the result among the solution obtained by Runge-Kuttaderiver single step method, Euler driver single step method and Adam Bashforth method with taking an example and they conclude that Adam Bashforth method gives accurate solution than the other methods as the number of step increases.

FaranakRabiei and Fudziah Ismail (2011) have published a paper in which they established the set of specific modified Runge-Kutta methods. Performance and exactness of modified RK method is presented and stability region is also discussed. They concluded the modified 3rd order RK method has less number(almost two step) of evaluation than the Butcher’s classical Runge-Kutta method of third order while this improved method preserving the same order of approximation and this new method is also useful for solving numerical integration of 1st order ordinary differential equations. He also concluded that IRK3 and IRK3-3 are almost two-stage method and both are computationally more efficient and gave less errors compared to the existing Runge-Kutta methods of the same order.

Third order method with two stages (IRK3)

= f()

= f()

= f()

= f()

and = + h( + ())

It is generated formula when the value of = 4/5

The formula when =1/2

= f()

= f()

= f()

= f()

and = + h( + ())

· Third order Improved Runge-kutta method with three stages(IRK3-3)

= f()

= f()

= f()

= f()

= f()

= f()

= + h( - () + ())

These are the three stage IRK methods when =1/3

Furthermore, IRK3-3 method when =1/2

= f()

= f()

= f()

= f()

= f()

= f()

= + h( + () + ())

Mohammed M. Ismail (2011) has proposed a new Runge-Kutta method of six-order which depends on the new fifth order Runge-Kutta method of David Goeken and also presented a comparison between this new method and new fifth order Runge-kutta method. He has concluded that the performance of this method is useful for the equation containing higher order derivative. And also presented a technique of applying a estimation to y’’.

Musa H et. al (2010)have brought down the simplification of the derivation of Runge-Kutta method and analysis of 4th order of this method by traversing some possibly conventional works. In this research paper he proposed each and every step of derivation and also plot the stoutness region.

David Goeken & Olin Johnson(1999) have addressed 3rd order and 4th order numerical integration method inspired by Runge-kutta method. In this paper they have reviewed these two methods and also presented a general solution of ordinary differential equation by using 5th order Runge-kutta method in 4 steps while standard Runge-Kutta methods requires 6. They proposed a method which is more efficient than standard Runge-Kutta method. It is more efficient where

1. and y’’ is cheaper to find than f.
2. Using an approximation of f’ reduces the step from 6 to 4 in fifth order case.

•  In Nov 1999, Dr John Butcher suggested a 5th order Runge-Kutta method by using the approach which based on tree. He has focused the main strands in the development of linear multistep method and Runge-Kutta method, both for general problems and also for several problems.
•  In 19th century, E.J Nystrom had ended the incomplete work of butcher. He completed the whole analysis of 5th order Runge-Kutta method and also introduced a new method for solving second order differential equation systems y’’(x) = f(y(x)), y() = , y’() =

It can be reduced in 1st order system

y’(x) = z(x), y() =

z’(x) = f(y), z() =

Now it can be easily solve by standard RK methods but he has concluded that, more efficient solution can obtain and less number of calculations can do if system of equation is solved in its original form rather than as converted form of first order.

References:

1. C.J Budd, A. Isercles, Geometric integration: numerical solution of differential equation on manifolds, philosophical transactions: mathematical, physical and engineering services 357(1999) 945-956.
2. Y. Z. Peng. Exact solutions from some nonlinear partial differential equations, physics Letters A 314(2003) 401-408.
3. R. V. Culshaw, S. Raun, A delay-differential equation model of HIV infection of CD4 + T-cells, Mathematical Biosciences 165(2000) 27-39.
4.  G.A Bocharov, F.A. Rihan numerical modelling in biosciences using delay differential equations, Journal of computational and applied mathematics 125(2000) 183-199.
5.  R. Norberg, Differential equation for moments of present values in life insurance, Insurance: mathematics and economics 17(1995) 171-180.
6. Ince, E.L., Ordinary Differential Equations, 1st Edition, 1956, Dover publications, Inc., New York.
7. Newton,I., The Mathematical Works, ed. D.T. Whiteside, 2 Vols, Johnson Reprint Corp., 1964-67.
8. Sasseer, Mathhematical origins of ordinary differential eqautions: The first hundred years (publishing year is not available)
9. Siddiqui and Manchanda, A First course in Differential equations with applications, Macmillan, 2006.
10. Shampine, L.F., and Watts, H.A. 1977, in Mathematical software III, J.R. Rice, ed. (New York; Academic Press), pp.257-275; 1979, Applied Mathematics and Computation, vol. 5, pp. 93-121.
11. F.Q. Zhang, Boundary value problems for second order differential equations with piecewise constant arguments, Ann. Diff. Eqs. (1993) 369–374.
12. R. Yuan, Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument, Nonlinear Anal. 41 (2000) 871–890.
13. J.J. Nieto, R. Rodrı´guez-Lopez, Green’s function for second-order periodic boundary value problems with piecewise constant arguments, J. Math.Anal.Appl. 304 (2005) 33–57.
14. P. Yang, Y. Liu, W. Ge, Green’s function for second order differential equations with piecewise constant arguments, Nonlinear Anal. Theory Meth. Appl.64 (8) (2006) 1812–1830.
15. L. Pan, Periodic solutions for higher order differential equations with deviating argument, J. Math. Anal. Appl. 343 (2) (2008) 904–918.
16. R. Rodrı´guez-Lopez, Nonlocal boundary value problems for second-order functional differential equations, Nonlinear Anal. 74 (2011) 7226–7239
17. Mohamed ,’etal’, Efficient two- derivative Runge-kutta-Nystrom methods for solving general second-order ordinary differential equations, Hindawi,2018.
18. Senthilnathan, A numerical solution of Initial value problems for Ordinary differential equations with Euler and Higher order of Runge-Kutta method using MATLAB, IJESI, VOLUME 7, Issue 4 Ver. III, April 2018.
19.  Claudio Faria Lopes Junior, Silva ed. Int. Journal of Engineering, Vol. 8, Issue 5(Part-II) May 2018, pp24-29.