SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS EXAMPLES

where the coefficients A1, A2, , Anare determined by solving the associated homoge-neous system of algebraic equations obtained by substituting expressions (12.6.2.9) intothe differential equations of the system in question and dividing by xσ. Since the system ishom[r]

15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 6892◦. At the second stage, we successively substitute the Φi(X)andΨj(Y ) of (15.5.1.4)into all solutions (15.5.4.1) to obtain systems of ordinary differential equations* for theunknown functi[r]

Any n × n hyperbolic system allows for n continuous solutions [of system (15.14.4.65)]corresponding to n characteristic velocities λ = λk. The continuous solutions are deter-mined by n systems of ordinary differential equations. Each system is represented by[r]

15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev´eTest,UsingTruncatedExpansions 75315.12. Methods of the Inverse Scattering Problem (Soliton Theory) . . . . . . . . . . . . . . . . . . . . 75515.12.1. MethodBasedonUsingLaxPairs 75515.12.2. M[r]

4).AcknowledgementsThis study was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant10971183.Authors’ contributionsAll authors contributed equally to the manuscript and typed, read and approved the final manuscript.Competing interestsThe aut[r]

subharmonic orbits,” Journal of D ifferential Equations, vol. 94, no. 2, pp. 315–339, 1991.7 S. Tersian and J. Chaparova, “Periodic and homoclinic solutions of extended Fisher-Kolmogorovequations,” Journal of Mathematical Analysis and Applications, vol. 260, no. 2, pp. 4[r]

In this paper, the Mawhin’s continuation theorem in the theory of coincidence degree has been used to investigate the existence of solutions for a class of nonlinear second-order differential systems of equations in ℝ

difficulties come into being in the investigation of the systems of degenerate ellipticequations. Some results for weakly related degenerate elliptic systems are obtained i n[5-7]. Particularly, these articles deal with Dirichlet type problems for the ellipticsystema(r)u[r]

[r]

Moscow, 1962.REFERENCES FOR CHAPTER 12 551Keller, H. B., Numerical Solutions of Two Point Boundary Value Problems, Society for Industrial & AppliedMathematics, Philadelphia, 1976.Kevorkian, J. and Cole, J. D., Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York[r]

pa–1(p + 1)b–1dp. (12.4.2.18)Remark 1. If a is noninteger, it is necessary to separate the real and imaginary parts in (12.4.2.18) toobtain real solutions.Remark 2. By setting α =–∞ and β = 0 in (12.4.2.17), one can find a second solution to equation(12.4.2.15) (at least for x > 0).12.4.3. Asy[r]

}free_vector(fx,1,iest);}CITED REFERENCES AND FURTHER READING:Stoer, J., and Bulirsch, R. 1980,Introduction to Numerical Analysis(New York: Springer-Verlag),§7.2.14. [1]Gear, C.W. 1971,Numerical Initial Value Problems in Ordinary Differential Equations(EnglewoodCliffs, NJ: Prent[r]

Table 6. Total times of Red - Black SOR method and Jacobi method.N 60 120 180 240 300SOR(seconds) 1 2 7 12 19Jacobi (seconds) 4 45 200 720The Red - Black SOR method is clearly the fastest one in terms of serial time and the numberof iterations.Table 1,3,4 show that when the number o[r]

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722Chapter 16. Integration of Ordinary Differential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.[r]

and the remainder at xf.This chapter will consider exclusively the initial value problem, deferring two-point boundary value problems, which are generally more difficult, to Chapter 17.The underlying idea of any routine for solving the initial value problem isalways this: Rewrite the dy’s and[r]

5. Bihari, IA: A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation.Acta Math Acad Sci Hung. 7,81–94 (1956). doi:10.1007/BF020229676. Cheung, WS: Some discrete nonlinear inequalities and applications to boundary value p[r]

Longstaff, F. A. and E. S. Schwartz (2001) Valuing American options by simulation: asimple least-squares approach. Review of Financial Studies, 14:113–147.Lowenstein, Roger (2001) When Genius Failed. London: Fourth Estate.Madan, Dilip B. (2001) On the modelling of option prices. Quanti[r]

y = –2–23Figure 31.9◆EXAMPLE 31.17 An industrial plant produces radioactive material at a constant rate of 4 kilograms per year.The radioactive material decays at a rate proportional to the amount present and has a half-life of 20 years.(a) Write a differential equation whose so[r]

4 R. D. Nussbaum, “Uniqueness and nonuniqueness for periodic solutions of xt−gxt − 1,”Journal of Differential Equations, vol. 34, no. 1, pp. 25–54, 1979.5 P. Dormayer, “The stability of special symmetric solutions of ˙xtαfxt − 1 with smallamplitud[r]

over a Rlemanmian manifold X, where
Gu(œ, 1) = / g(,)u(y)4
1s an integral operator, (4œ, ý) 1s a differential form on Ã, A —= 2(đ-Eỏ) is a natural differential
operator in X. We consider the case when X is a tore II. It is shown that th[r]