Riccati Equation: Numerical Solutions via Adam-Bashforth & Moulton

This project investigates the numerical solutions of Riccati differential equations using the Adam-Bashforth and Adam-Moulton multistep methods. The study begins by introducing the theoretical background of differential equations, highlighting the limitations of analytical solutions and the importance of numerical approximations. It describes the core concepts of single-step methods, such as Euler and Runge-Kutta, and contrasts them with the multistep approaches, particularly Adam-Bashforth (explicit) and Adam-Moulton (implicit) methods. The project then details the methodology, including the derivation of the methods and the application of Maple and Matlab for computation and analysis. The literature review explores the existing research on the application of these methods to solve differential equations, specifically Riccati equations. The findings indicate that while both methods provide accurate results, the Adam-Bashforth method demonstrates higher accuracy compared to the Adam-Moulton method. The project underscores the significance of multistep methods in reducing computational time and improving efficiency in solving differential equations, particularly in cases where analytical solutions are not feasible.

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-
tial equations.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 Riccati differentialequation 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
1 Introduction
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
of the 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)
1.1 Problem 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.
1.2 Significant 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.
1.3 Scope 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.
2 Literature 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) of ordinary 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
initial value problem.As from Areo & Adeniyi(2013) ,the solution is in the
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form of first derivatives of y in terms of x and y.There is range for x which
is [a,b] and it’s a finite real number.Therefore the solution is the form of first
derivative before a unique solution.For general linear multistep method of step
k, the derivatives y only need to be evaluate a few times which is less than the
number of evaluations for the one-step method in the range of integration [a,b].
So by Butcher (2000) it is proof that the linear multistep method is better than
one-step method.
Regarding Anake & Ashibel Cugp (2011) one step methods include the Eu-
ler’s methods, the Runge-Kutta methods and the theta methods.These meth-
ods are only suitable for the solutions of first order IVPs of ODEs because of
their very low order of accuracy which say by C Butcher John (1987).However,
by Petzoldf (1986) in order to develop higher order one step methods such as
Runge-Kutta methods, the efficiency of Euler methods in terms of the number
of function evaluations per step is sacrificed since more function evaluations is
required.
On the other hand oflinear multistep methods,include methods such as
Adam-Bashforth method, Adam-Moulton method, and Numerov method.These
methods give high order of accuracy and are suitable without necessarily reduc-
ing it to an equivalent system offirst order IVPs ofODEs. Linear multistep
methods are not as efficient,in terms of function evaluations,as the one step
method and also require some values to start the integration process.
Various method are used to solve the Riccati differential equation and of it
is Bezier curves.In that method, the Bezier polynomial of degree n is develop
and new efficient method, the multistage variational iteration is applied.File &
Aga (2016) said that many authors are working with the Bezier curves.Some
of them use Bezier control point in approximating date and function while some
are using it to solve differential equation numerically.
In addition,solving delay differentialequation and singular perturbed two
points boundary value problems are also using Bezier curves.Amman & Neudecker
(1997) pointed out that solving the riccati equation recursively in time is simple
which generally doesn’t pose any difficulties.
Finally, one of classin nonlinearequation are Riccatidifferentialequa-
tion and have played many roles in applied science.One-dimensionalstatic
Schr¨odinger equation is closely related to the Riccati differential equation.The
Riccati differential equation is named after the Italian nobleman Count Jacopo
Francesco Riccati(1676–1754).The applications of this equation maybe found
not only in random processes, optimal control, and diffusion problems but also
in stochastic realization theory, optimal control, network synthesis and financial
mathematics.
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3 Methodology
This section is divided into 4 subsections.In the first subsection, the introduc-
tion to Adam-Bashforth and Adam-Moulton Methods are stated and in the sec-
ond subsection, the introduction of Riccati Equation is then stated.In the third
subsection, a little explanation on how to derive a higher step Adam-Bashforth
and Adam-Moulton is discussed.Last but not least,the process ofanalyzing
the accuracy ofAdam-Bashforth and Adam-Moulton methods is discussed in
the final subsection.
3.1 Introduction of Adam-Bashforth and Adam-Moulton
Methods
3.1.1 Introduction of Adam Bashforth
Given a starting value problem like dy/dx = f (x, y), y(x0) = y0 together with
extra values y1 = y(x0+h), ..., yk−1 = y(x0+(k−1)h) an explicit linear multistep
method that approximates the solution is the Adams-Bashforth k-step method
is, y(x) at x = x0 + kh, of the initial value problem
yk = yk−1 + h(a0f (xk−1 , yk−1 ) + a1f (xk−2 , yk−2 ) + ... + ak−1 f (x0, y0))
where c0, c1, ..., ck−1 are constants.
The constants ai can be known by assume that an exact for polynomials in
x of degree k − 1 or less, by the order of the Adams-Bashforth method is k.The
major strength of the Adams-Bashforth method over the Runge-Kutta methods
is the integral f(x,y) can be evaluating by one evaluation at each step
3.1.2 Introduction of Adam-Moulton
Adam-Moulton method (k − 1) is linear multistep or more specific an implicit
method that mostly approximates the solution y(x) at x = x0 + kh,of IVP as
yk = yk−1 + h(b0f (xk , yk ) + b1f (xk−1 , yk−1 ) + ... + bk − 1f (x1, y1))
where d1, ..., dk−1 are constants.
Assuming that the linear expression is exact for polynomials as the constants
bi should be determined in x ofdegree k − 1 or less,and another case ofthe
Adams-Moulton is k
Adam-Bashforth method is used to generate an initialestimate for yk for
starting the Adams-Moulton iterative method.Then Adams-Moulton formula
is then used to generate successive estimates for yk . Providing that the step
size h need to process of converges is choose so that |hf, y(x, y)|< 1 over the
region, where f ,y acts as the partial derivative of f with respect to y.
The predictor is been forms by Adams-Bashforth method while the corrector
that been form by Adams-Moulton method,then by given historicalvalues a
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predictor-corrector multistep procedure in approximating the solution ofany
differential equation.
Mostly a (k − 1)-step Adams-Moulton method is combine with a k-step
Adams-Bashforth method.However,it is not necessary to pair any 1-step Moul-
ton method with k-step Bashforth method.
3.2 Introduction to Riccati Equation
Differential equation are often used to model complex problems in science and
engineering.In most practical problem,this equations are highly nonlinear and
are too complicated to solve analytically.However,by given initialcondition
these differentialequation can be solved approximately using numericalinte-
gration methods.Riccati equation is chosen in order to explain how Adam-
Bashforth and Adam-Moulton methods worked.Show below is a discrete form
of Riccati equation
dy
dx = A(x)y2 + B(x)y + C(x)
Since v(x) = y(x) − y1(x), we have
y(x) = v(x) + y1(x)
Then differentiate y on x ,
y0
(x) = v0
(x) + y0
1(x)
Suppose y1(x) is a solution Riccati equation
Let v(x) = y(x) − y1(x) be a variable and transform this equation into
Bernoulli equation,
Since y1(x) solves the riccati equation,
y0
1 = A(x)y2 + B(x)y + C(x)
Substitute into equation,
v0+ y0
1 = A(x)[v + y1]2 + B(x)[v + y1] + C(x)
Then,
v0
+[A(x)y2+B(x)y+C(x)] = A(x)v2+2A(x)y1v+A(x)y2
1+B(x)v+B(x)y1+C(x)
v0 = A(x)v2 + 2A(x)y1v + B(x)v
v0+ [−2A(x)y1 − B(x)]v = A(x)v2
This is in the form of a Bernoulli equation.
where,
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p(x) = −2A(x)y1 − B(x)
q(x) = A(x)v2
3.3 Accuracy of Analysis
The result of analysis for both Adam-Bashforth and Adam-Moulton are been
compared with the exact solution.To show the accuracy both methods,the
absolute of error is used.Below show the equation to find the absolute error.
||~e|| =
v
u
u
t kX
n=1
(X − X 0)2
3.4 Analysis of apply both Adam-Bashforth and Moulton
methods
Applying 4-steps of Adam-Bashforth and Adam-Moulton methods into Riccati
equation.The calculation is show in MATLAB software and allthe result is
presenting in appendix.
4 Results And Discussion
In this section, all the result obtained from the calculation are presented.This
part contained the explanation and elaboration ofthe obtained result.The
interpretation of the formula has been shown in implementation section.
4.1 Result
Figure below shows the results of accuracy of the two methods.
Table 1:Result Value
x-value Exact Solution Adam-Bashforth ApproximationAdam-Moulton Approximation
0 0.5 0.5 0.5
0.1 0.47502081252106 0.475020813824448 0.475020813824448
0.2 0.450166002687522 0.450166005329146 0.450166005329146
0.3 0.425557483188341 0.425557487247527 0.425557487247527
0.4 0.401312339887548 0.401313141996514 0.401312348052676
0.5 0.377540668798145 0.377542185792827 0.377540682469658
0.6 0.354343693774205 0.354345816611707 0.354343713954574
0.7 0.331812227831834 0.331814828823113 0.331812255073873
0.8 0.310025518872388 0.310028459697363 0.310025553267906
0.9 0.289050497374996 0.289053637143753 0.289050538581219
1.0 0.268941421369995 0.268944624062497 0.26894146866699
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Table 2:Comparison of Error Analysis
x-value Adam-Bashforth Error Adam-Moulton Error
0 0 0
0.1 1.30338767556637e-091.30338767556637e-09
0.2 2.6416240306304e-092.6416240306304e-09
0.3 4.05918604284849e-094.05918604284849e-09
0.4 8.02108966246884e-078.16512846224526e-09
0.5 1.51699468120547e-061.36715126997089e-08
0.6 2.12283750217201e-062.0180369930678e-08
0.7 2.60099127902258e-062.72420386249195e-08
0.8 2.94082497581671e-063.43955184334277e-08
0.9 3.13976875654376e-064.12062228227761e-08
1.0 3.20269250203564e-064.72969950982005e-08
Table 2 show the comparison between Adam-Bashforth and Adam-Moulton
methods.As a final result show that Adam-Moulton is much better in accuracy
with less an error.This is because Adam-Moulton use technique with one-step
ahead than Adam-Bashforth to compute the better result.
4.2 Discussion
After the result obtained,it show that approximation ofAdam-Moulton on
Riccati equation is much smaller than Adam-Bashforth.The accuracy of anal-
ysis is discuss from the result show that Adam-Moulton is more accurate than
Adam-Bashforth.As my opinion,the RiccatiEquation is an example of non-
linear differential equation that can be solve using Adam-Bashforth and Moulton
methods.Regarding the result, both Adam-Bashforth and Moulton only can be
used in first order of differentialequation.Thus, it is relevant to approximate
Riccati Equation using both methods and compare with analytical result.
Advantages ofthis methods is stable even ifsmall changes in the initial
condition result in only small changes in the computed solution.Moreover this
method requires only one new function evaluation for each step.It will lead
to great savings in time.Disadvantages of this method is that it require four
function evaluation from RK4 method to solve.It means that this method is
depend on the other method.
5 Conclusion
In this study, Adam-Bashforth and Adam-Moulton methods was applied to solve
Riccatiequation.Both method was derived to get 4-step for Adam-Bashforth
and Adam-Moulton using manualcalculation and continue using Maple soft-
ware. Hence both methods are used to solve nonlinear differentialequation
such as Riccati equation.The Riccati equation are obtained from Ghomanjani
& Khorram (2017) to help in solving using Adam-Bashforth and Adam-Moulton
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methods.After that, the accuracy of both Adam methods on the Riccati equa-
tion were analysed.From this analysis, Adam-Bashforth method was found to
be more accurate than Adam-Bashforth methods.
After finish this study,there are many numericalmethods been used to
solve any differential equation.In future, these methods can be explore whether
suitable to used on any other non-linear differential equation.
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