APPLICATIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN COMPUTER SCIENCE

where J0(x) is the Bessel function of the first kind and I0(x) is the modified Bessel function of the first kind.In this case, only one solution (12.4.2.10) has been obtained. This is due to the fact that the other solutiongoes to infinity as x → 0, and hence formula (12.4.2.6) cann[r]

Table 2.N 24 30 40Toltal times(seconds) 120 180 300For better convergence, we use other methods, such as the parallel Jacobi method [5], the parallelSOR Red/Black [6,7,8,9] method.The parallel Jacobi method and Parallel SOR Red/Black method are implemented in C and MPIand executed on 1 node <[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 aphase[r]

¨Unal and A. Zafer, “Oscillation of second-order mixed-nonlinear delay dynamic equations,”Advances in Difference Equations, vol. 2010, Article ID 389109, 21 pages, 2010.16 Z. Zheng, X. Wang, and H. Han, “Oscillation criteria for forced second order differential equationsw[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[r]

.(12.7.2.4)If the case a = b = c, where system (12.7.2.2) can be integrated directly, does not take place, the above twofirst integrals (12.7.2.3) and (12.7.2.4) are independent. Hence, using them, one can express y2and y3in termsof y1and then substitute the resulting expressions into the first equati[r]

x, C1)=0, G(y, ψy)=C1.On solving these equations for the derivatives, we obtain linear separable equations, whichare easy to integrate.References for Chapter T7Kamke, E., Differentialgleichungen: L¨osungsmethoden und L¨osungen, II, Partielle DifferentialgleichungenErster Ordnung f¨ur[r]

Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2011, Article ID 979586, 10 pagesdoi:10.1155/2011/979586Research ArticleFixed-Point Results for Generalized Contractionson Ordered Gauge Spaces with ApplicationsCristian Chifu and Gabriela Petrus¸elFaculty of Business, Ba[r]

tions, vol. 335, no. 2, pp. 1052–1060, 2007.[15] J. R. L. Webb, “Positive solutions of some three point boundary value problems via fixed pointindex theory,” Nonlinear Analysis: Theory, Methods &amp; Applications, vol. 47, no. 7, pp. 4319–4332,2001.[16] R. P. Agarwal, D. O’Regan, an[r]

446 References117. B. Lewis and D.J. Berg. Multithreaded Programming with Pthreads. Prentice Hall, NewJersey, 1998.118. J.W.H. Liu. The role of elimination trees in sparse factorization. The SIAM Journal on MatrixAnalysis and Applications, 11: 134–172, 1990.119. D.T. Marr, F. Bi[r]

78. J.L. Gustafson. Reevaluating Amdahl’s law. Communications of the ACM, 31(5): 532–533,1988.79. W. Hackbusch. Iterative Solution of Large Sparse Systems of Equations. Springer, New York,1994.80. K. Hammond and G. Michaelson, editors. Research Directions in Parall[r]

x→0+( f (x)/x), g0:= limx→0+(g(x)/x), f∞:= limx→∞( f (x)/x), andg∞:= limx→∞(g(x)/x) exist as real numbers.There is an ongoing flurry of research activities devoted to positive solutions of dy-namic equations on time scales (see, e.g., [1–7]). This work entails an extension of<[r]

gt, xt − ks, 1.2where x ∈ Rp, f ∈ CRp, Rp,andg ∈ CR × Rp, Rp,andr&gt;0,s &gt;0 are two given constants.2 Journal of Inequalities and ApplicationsFor the special case that n  1andp  1, various problems on the solutions of 1.1,such as the existence of periodi[r]

Keeping the U.S. Computer Industry Competitive: Systems Integration (Free Executive Summary)http://www.nap.edu/catalog/1914.htmlFree Executive SummaryISBN: 978-0-309-04544-5, 106 pages, 6 x 9, paperback (1992)This executive summary plus thousands more available at www.nap.edu.Keeping the U.S.[r]

(t) is continuous on [α, β], too. Then, by a similarmethod to the proof of (2.3) together with Lemma 2.2, we can obtain (2.4) immediately.For the other ordinary cases, i.e., a = 0, we only need to move the interval [α, β] evenlysuch that this interval symmetrizes about the origin. The[r]

matically, and do not seem to be readily digestible in prac-tical applications (Boxler [37]). Intuitively, however, the linkbetween the center manifold theory and stochastic dynam-ics seems to be quite natural. As shown above, under certainconditions, variance of fluctuations aro[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[r]

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

Security+Brian E. BrzezickiAbout MeInstructorBrian E. Brzezickiemail: brianb@paladingrp.comBachelor of Science, Computer ScienceMasters of Science, Computer ScienceISC2 CISSPEC-Council Certified Ethical Hacker (CEH)CompTIA Security+Red Hat Certified Technici[r]

15.6.1-2. Solution by reduction to equations with quadratic (or power) nonlinearities.In some cases, solutions of the form (15.6.1.1) can be searched for in two stages. First, onelooks for a transformation that would reduce the original equation to an equation with aquadr[r]