Numerical Solutions of Riccati Equation PDF

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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
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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
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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
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