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Numerical Solutions of Riccati Equations Using

Adam-Bashforth and Adam-Moulton Methods

Farahanie Fauzi

Mohamad Nazri Mohamad Khata

Nur Habibah Radzali

Mohamad Aliff Afifuddin Hilmy

Abstract

A differentialequation can be solved analytically or numerically.In

many complicated cases,it is enough to just approximate the solution if

the differentialequation cannot be solved analytically.Euler’s method,

the improved Euler’s method and Runge-Kutta methods are examples of

commonly used numericaltechniques in approximately solved differen-

tialequations.These methods are also called as single-step methods or

starting methods because they use the value from one starting step to

approximate the solution of the next step.While, multistep or continuing

methods such as Adam-Bashforth and Adam-Moulton methods use the

values from several computed steps to approximate the value of the next

step.So, in terms of minimizing the calculating time in solving differen-

tial,multistep method is recommended by previous researchers.In this

project,a Riccatidifferentialequation is solved using the two multistep

methods in order to analyze the accuracy of both methods.Both meth-

ods give small errors when they are compared to the exact solution but it

is identified that Adam-Bashforth method is more accurate than Adam-

Moulton method.

Keywords:ODE, Adam-Bashforth, Adam-Moulton, Riccati

1Introduction

It has been shown that a solution of a differential equation exist.But in many in-

stances, it is enough to just approximate the solution if the differential equation

cannot be solved analytically.Euler’s method, the improved Euler’s method and

Runge-Kutta methods are examples of commonly used numerical techniques in

approximately solved differentialequations.These methods are also called as

single-step methods or starting methods because they use the value from one

starting step to approximate the solution of the next step.In the other hand,

multistep or continuing methods such as Adam-Bashforth and Adam-Moulton

1

Adam-Bashforth and Adam-Moulton Methods

Farahanie Fauzi

Mohamad Nazri Mohamad Khata

Nur Habibah Radzali

Mohamad Aliff Afifuddin Hilmy

Abstract

A differentialequation can be solved analytically or numerically.In

many complicated cases,it is enough to just approximate the solution if

the differentialequation cannot be solved analytically.Euler’s method,

the improved Euler’s method and Runge-Kutta methods are examples of

commonly used numericaltechniques in approximately solved differen-

tialequations.These methods are also called as single-step methods or

starting methods because they use the value from one starting step to

approximate the solution of the next step.While, multistep or continuing

methods such as Adam-Bashforth and Adam-Moulton methods use the

values from several computed steps to approximate the value of the next

step.So, in terms of minimizing the calculating time in solving differen-

tial,multistep method is recommended by previous researchers.In this

project,a Riccatidifferentialequation is solved using the two multistep

methods in order to analyze the accuracy of both methods.Both meth-

ods give small errors when they are compared to the exact solution but it

is identified that Adam-Bashforth method is more accurate than Adam-

Moulton method.

Keywords:ODE, Adam-Bashforth, Adam-Moulton, Riccati

1Introduction

It has been shown that a solution of a differential equation exist.But in many in-

stances, it is enough to just approximate the solution if the differential equation

cannot be solved analytically.Euler’s method, the improved Euler’s method and

Runge-Kutta methods are examples of commonly used numerical techniques in

approximately solved differentialequations.These methods are also called as

single-step methods or starting methods because they use the value from one

starting step to approximate the solution of the next step.In the other hand,

multistep or continuing methods such as Adam-Bashforth and Adam-Moulton

1

methods use the values from several computed steps to approximate the value

of the next step.

Since linear multistep methods need several starting values to compute the

next value, it is necessary to use a one step method to compute enough starting

values of the solution to be able to use the multistep method.

First-order numericalprocedure for solving ordinary differentialequations

(ODEs) like Euler method with a given initialvalue.Simplest Runge–Kutta

method is the custom of basic explicit method for numericalintegration in an

ordinary differential equations.Euler method refers to only one previous point

and its derivative to determine the current value.A simple modification of the

Euler method which eliminates the stability problems is the backward Euler

method.This modification leads to a family of Runge-Kutta.

Runge–Kutta methods are a family of implicit and explicit iterative methods,

which includes the well-known routine called the Euler Method.The most

popular and widely used is RK4 because its less computationalrequirement

and high accuracy.This RK4 is an example of one-step method in numerical,

Petzoldf(1986).Development ofmodified this RK4 leads from one-step to

multi-step method,like Adam’s methods.

Adam-Bashforth method and Adam-Moulton methods are the families of

linear multistep method that commonly used.Adam-Bashforth methods is an

example of explicit methods of multi-step,Garrappa (2009).Adam Bashforth

method are designed by John Couch Adams to solve a differentialequation

modeling capillary action due to Francis Bashforth, Misirli & Gurefe (2011)

While Adam-Moulton methods is an example ofimplicit methods.The

backward Euler method can also be seen as a linear multistep method with one

step.It is the first method of the family of Adams–Moulton methods, and also

ofthe family ofbackward differentiation formulas.Adam-Moulton methods

are solely due to John Couch Adam,just like Adam-Bashforth method.The

name of Forest Ray Moulton become associated with these methods because he

realized that they could be used in tandem with Adam-Bashforth Method as a

predictor-corrector pair.Jator (2001)

Non-linear differentialequation are commonly used in spring mass system,

resistor capacitor induction and many more.A part of this non-linear is Riccati

differential equation which is well-known among them.This equation is named

after Jacopo Francesco Riccati.Solution of Riccati equation is usually solved by

two numerical technique which are cubic B-spline scaling function and Cheby-

shev cardinalfunction and also used to refer to matrix equation are shown in

File & Aga (2016).Riccati equation play a fundamental role in financial math-

ematics, network synthesis and optimal control.Ghomanjani & Khorram (2017)

1.1Problem Statement

Basically, single-step methods especially Runge-Kutta method is often used be-

cause ofits accuracy.However,the process ofcalculation is time consuming

since the differential equation need to be evaluated several times at every step.

For example, the Runge-Kutta of order 4 (RK4) method requires four functions

2

of the next step.

Since linear multistep methods need several starting values to compute the

next value, it is necessary to use a one step method to compute enough starting

values of the solution to be able to use the multistep method.

First-order numericalprocedure for solving ordinary differentialequations

(ODEs) like Euler method with a given initialvalue.Simplest Runge–Kutta

method is the custom of basic explicit method for numericalintegration in an

ordinary differential equations.Euler method refers to only one previous point

and its derivative to determine the current value.A simple modification of the

Euler method which eliminates the stability problems is the backward Euler

method.This modification leads to a family of Runge-Kutta.

Runge–Kutta methods are a family of implicit and explicit iterative methods,

which includes the well-known routine called the Euler Method.The most

popular and widely used is RK4 because its less computationalrequirement

and high accuracy.This RK4 is an example of one-step method in numerical,

Petzoldf(1986).Development ofmodified this RK4 leads from one-step to

multi-step method,like Adam’s methods.

Adam-Bashforth method and Adam-Moulton methods are the families of

linear multistep method that commonly used.Adam-Bashforth methods is an

example of explicit methods of multi-step,Garrappa (2009).Adam Bashforth

method are designed by John Couch Adams to solve a differentialequation

modeling capillary action due to Francis Bashforth, Misirli & Gurefe (2011)

While Adam-Moulton methods is an example ofimplicit methods.The

backward Euler method can also be seen as a linear multistep method with one

step.It is the first method of the family of Adams–Moulton methods, and also

ofthe family ofbackward differentiation formulas.Adam-Moulton methods

are solely due to John Couch Adam,just like Adam-Bashforth method.The

name of Forest Ray Moulton become associated with these methods because he

realized that they could be used in tandem with Adam-Bashforth Method as a

predictor-corrector pair.Jator (2001)

Non-linear differentialequation are commonly used in spring mass system,

resistor capacitor induction and many more.A part of this non-linear is Riccati

differential equation which is well-known among them.This equation is named

after Jacopo Francesco Riccati.Solution of Riccati equation is usually solved by

two numerical technique which are cubic B-spline scaling function and Cheby-

shev cardinalfunction and also used to refer to matrix equation are shown in

File & Aga (2016).Riccati equation play a fundamental role in financial math-

ematics, network synthesis and optimal control.Ghomanjani & Khorram (2017)

1.1Problem Statement

Basically, single-step methods especially Runge-Kutta method is often used be-

cause ofits accuracy.However,the process ofcalculation is time consuming

since the differential equation need to be evaluated several times at every step.

For example, the Runge-Kutta of order 4 (RK4) method requires four functions

2

evaluation for every step.Otherwise,a multistep method need only one new

function to evaluate for every step.So, it is best to apply multistep method to

solve differential equations in order to reduce the time required in the calculation

process.

1.2Significant Of Project

This topic is chosen because some people only know how to approximate value

using basis methods.By using this linear multi-step method, other mathemati-

cians will understand that there a better and easier way to approximate a value.

Furthermore, it will inspire new mathematicians to invent new formula that can

be derived from an old formula.

1.3Scope Of Project

This project focused on solving Riccati differential equations by using numerical

methods which are Adam-Bashforth and Adam-Moulton method.Deriving the

4-step of both methods require Maple application while the final result of Riccati

equation require Matlab application.The combination ofboth applications

provide easier way to solve the Riccati differential equations.

2Literature Review

Adam-Bashforth and Adam-Moulton are explicit/implicit numericalintegra-

tion.Both methods can solve as an approximation in nonlinear differential

equation.TraditionalAdam-Bashforth-Moulton predictor-corrector method is

proposed long ago and since then the methods have been continuously improved.

Adam Bashforth was derived explicitly using Newton Backward Difference

Formula with an equal of spacing points.In order to differenciate Adam Bash-

forth and Adam Moulton Method,the mathematician proposed the use of m-

step for Adam Bashforth and m-1 for Adam Moulton.As a conclusion,both

method are already derived by Chiou & Wu (1999).

Then according Aboiyar et al.(2015) solving first order initialvalue prob-

lems (IVPs) ofordinary differentialequation with step number m=3.This

journal using Hermite polynomials as basis function.Using the collocation and

interpolation technique Adam-Bashforth,Adam-Moulton and OptimalOrder

Method was invented.Then to derive three step of Adam-Bashforth is set n=3

and Adam Moulton is sets n=4 in equation probabilist’s Hermite polynomial.

As a conclusion, the best result was obtained and been compared to see which

method give the best approximation with less of error.

Furthermore,a direct solution can be developed by using Adam-Moulton

methods comes from Jator (2001).This solution must be used to calculate the

initialvalue problem.As from Areo & Adeniyi(2013) ,the solution is in the

3

function to evaluate for every step.So, it is best to apply multistep method to

solve differential equations in order to reduce the time required in the calculation

process.

1.2Significant Of Project

This topic is chosen because some people only know how to approximate value

using basis methods.By using this linear multi-step method, other mathemati-

cians will understand that there a better and easier way to approximate a value.

Furthermore, it will inspire new mathematicians to invent new formula that can

be derived from an old formula.

1.3Scope Of Project

This project focused on solving Riccati differential equations by using numerical

methods which are Adam-Bashforth and Adam-Moulton method.Deriving the

4-step of both methods require Maple application while the final result of Riccati

equation require Matlab application.The combination ofboth applications

provide easier way to solve the Riccati differential equations.

2Literature Review

Adam-Bashforth and Adam-Moulton are explicit/implicit numericalintegra-

tion.Both methods can solve as an approximation in nonlinear differential

equation.TraditionalAdam-Bashforth-Moulton predictor-corrector method is

proposed long ago and since then the methods have been continuously improved.

Adam Bashforth was derived explicitly using Newton Backward Difference

Formula with an equal of spacing points.In order to differenciate Adam Bash-

forth and Adam Moulton Method,the mathematician proposed the use of m-

step for Adam Bashforth and m-1 for Adam Moulton.As a conclusion,both

method are already derived by Chiou & Wu (1999).

Then according Aboiyar et al.(2015) solving first order initialvalue prob-

lems (IVPs) ofordinary differentialequation with step number m=3.This

journal using Hermite polynomials as basis function.Using the collocation and

interpolation technique Adam-Bashforth,Adam-Moulton and OptimalOrder

Method was invented.Then to derive three step of Adam-Bashforth is set n=3

and Adam Moulton is sets n=4 in equation probabilist’s Hermite polynomial.

As a conclusion, the best result was obtained and been compared to see which

method give the best approximation with less of error.

Furthermore,a direct solution can be developed by using Adam-Moulton

methods comes from Jator (2001).This solution must be used to calculate the

initialvalue problem.As from Areo & Adeniyi(2013) ,the solution is in the

3

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