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A Phase-fitted Numerov-type Method For Solving ODE with

Oscillatory Solution

NAME : ADIBAH BINTI MAT KHAIRULDINMATRIC NO : 156734 SUPERVISOR : DR NORAZAK SENU

OUTLINE

Introduction Objective

Literature review Methodology

References

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INTRODUCTION

Introduced by Boris Vasil’evich Numerov.

The term phase lag was first introduced by Brusa and Nigro in 1980.

Use to solve problems such as oscillatory problem, radial schrodinger equation and periodic initial value problem.

Boris Vasil’evich Numerov

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Numerov-type Method

Numerov-type method is a numerical method to solve ordinary differential equation of second order

with initial conditions

.

Note that the integrand does not depend of the first derivative .

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[1]

[2]

The dispersion and dissipation of the method are the errors of the numerical solution in phase and in amplification.

From Simos and Williams (2001), the quantity

, where and is the step length, and are polynomials in , is called the dispersion.

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[3]

The quantity, ,

where and is the step length, is polynomial in is called dissipation.

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[4]

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PROBLEM STATEMENT

The solution of the second order differential equation

with initial conditions

.

Solution is in oscillatory form.

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[5]

[6]

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OBJECTIVES

To develop explicit Numerov-type method for

oscillatory solution.

To compare the numerical result of new

phase-fitted Numerov-type method with

original Numerov-type method proposed by

Simos and Williams (2001).

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LITERATURE REVIEW

Chawla and Rao (1984)

Present a new Numerov-type fourth order method for which frequency distortion or phase-lag is a minimal using implicit method.

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Chawla and Rao (1986)

Proposed explicit modifications of the Numerov-type method trying simultaneously to increase the phase-lag order or other characteristics with algebraic orders six.

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Simos and Raptis (1990)

Develop Numerov-type methods with phase-lag of order eight and ten which are P-stable or have larger intervals of periodicity.

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Simos (1999)

Develop a family of explicit non-symmetric Numerov-type methods of algebraic order six and with phase-lag of order 10.

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Simos and Williams (2001)

Develop explicit Numerov-type methods with minimal phase-lag of algebraic order five and have phase-lag order eight and ten.

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Tsitouras and Simos (2002)

Introduce an embedded Numerov-type method of algebraic order eight with phase lag orders 10–20 for the numerical integration of second-order periodic initial value problems with ten stages.

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Bratsos et al. (2007)

Investigate the class of the initial value problems of second order with oscillatory solutions.

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Fang et al. (2012)

Investigate explicit Numerov-type methods for the numerical integration of periodic initial value problems(IVPs) with two frequencies.

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METHODOLOGY

1. Substitute scalar test equation from Lambert and Watson (1976) ,

into family of explicit fifth algebraic order method from Simos and Williams (2001).

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[7]

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Family of explicit fifth algebraic order method from Simos and Williams (2001) :

• ,• ,• ,

[8][9]

[15][14][13][12][11][10]

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,

where , and .

[17][16][18][19]

2. Apply a non-symmetric two step method to the scalar test to obtain a difference equation

where and is the step length, and are polynomials in .

3. The characteristic equation associated with equation [20] is

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[21]

[20]

4. Let two free parameter and substitute in the definition of phase-fitted in equation [3] and [4].

5. Solve for free parameter by using maple. Then, have new solution for free parameter

6. Run the new method and original method in the Microsoft Visual C++ program to compare the numerical result.

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REFERENCES

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Bratsos, A.G., Famelis, I.Th., Kollias, A. and Tsitouras, Ch.

2007. “Phase-fitted Numerov type methods”, Applied Mathematics and Computation, vol. 184 pp. 23–29.

Chawla, M.M. and Rao, P.S. 1984. “Numerov type method with minimal phase lag for the integration of second order periodic initial value problems”, Journal of Computational and Applied Mathematics, vol. 11 pp. 277-281.

Chawla, M.M. and Rao, P.S. 1986. “A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II- explicit method”, Journal of Computational and Applied Mathematics, vol. 15 pp. 329-337.

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Fang, X., You, X. and Chen, Z. 2012. “New Phase Fitted and Amplification Fitted Numerov-Type Methods for Periodic IVPs with Two Frequencies”, Abstract and Applied, Vol. 2012, pp. 15.

Lambert, J.D., Watson, I.A. 1976. “Symmetric multistep methods for periodic initial value problems”, Institute Mathematical Applied. vol. 18, pp. 189-202.

Simos, T.E. 1999. “Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation”, Computers and Chemistry vol. 23, pp. 439-446.

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Simos, T.E. and Raptis, A.D. 1990. ”Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrödinger equation”, Computing, vol. 45, pp. 175-181.

Simos, T.E. and Williams, P.S. 2001. “New insights in the development of numerov methods with minimal phase-lag for the numerical solution of the Schrodinger equation”, Computers and Chemistry vol. 25, pp. 77–82.

Tsitouras, Ch. and Simos, T.E. 2002. “High algebraic, high phase-lag order embedded Numerov-type methods for oscillatory problems”, Applied Mathematics and Computation, vol. 131 pp. 201–211.

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