SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS MATHEMATICA

,∂2w∂y2F =0= 0,(15.9.1.3)which coincides with equation (15.8.2.3).All three equations (15.9.1.1)–(15.9.1.3) are used for the construction of exact solutionsof the original equation (15.9.1.1). It should be observed that in this case, the determin-ing equations obtained for the unknown functions[r]

Modeling charge carrier distribution in low-doped zones of bipolar power semiconductor devices is known as one of the most important issues for accurate description of the dynamic behavior of these devices. The charge carrier distribution can be obtained solving the Ambipolar Diffusion Equation (ADE[r]

Equation (15.8.3.3) admits the operator X3= y∂x–x∂y(see Example 1 in Subsection 15.8.2), which definesrotation in the plane. The corresponding transformation is given in Table 15.7. Replacing x in (15.8.3.4) by ¯x(from Table 15.7), we obtain a one-parameter solution of equation (15.8.3.3):w =ln2(x co[r]

NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF OR[r]

can be found in Section 11.2. The Laplace transforms of some functions are listed inSection T3.1. Tables of inverse Laplace transforms are listed in Section T3.2. Such tablesare convenient to use in solving linear problems for partial differential equations.14.5.1-2. Solution procedure for linear pr[r]

The infinite system of differential equations for the nonequilibrium Green functions of electrons in a single-level quantum dot connected with two conducting leads is truncated by applying the mean-field approximation to the mean values of the products of four operators. As the result the system of[r]

xiv CONTENTS15.6.3. DifferentiationMethod 70015.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations andTheirApplications 70415.7. Direct Method of Symmetry Reductions of Nonlinear Equations . . . . . . . . . . . . . . . . . . 70815.7.1. Clarkson–Kruskal Direct Method . . . . . .[r]

Ổn định kích thích nhỏ và ứng dụng trên Phần mềm PSSE.NỘI DUNG CHÍNH PHẦN 21 (small signal stability and application of small signal stability): 1. Transient Stability: a. Time Domain Analysis. b. Step wise Integration of Differential Equations. 2. SmallSignal Stability. a. Frequency Domain An[r]

Interscience, New York, 1990.Kanwal, R. P., Generalized Functions. Theory and Technique, Academic Press, Orlando, 1983.Leis, R., Initial-Boundary Value Problems in Mathematical Physics, John Wiley & Sons, Chichester, 1986.Mikhlin, S. G. (Editor), Linear Equations of Mathematical Physics, Hol[r]

Sample 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. Permission is granted for internet users to make one paper copy for their own personal u[r]

order of the differential equation representing the differential constraint.For the types of solutions listed in Table 15.10, it is preferable to use the methods ofgeneralized and functional separation of variables, since these methods require less stepswhere it is necessary to solve intermediate di[r]

considered system (i.e., universal methods, applicable for any equation, are considered)[7].In the last years one can observe the development of the numerical methods which areapplicable for some clases of equations (e.g., admitting Hamiltonian formulation) butare much more powerful (especially when[r]

dydt=−4x−yyx31.6 SECOND ORDER HOMOGENEOUS DIFFERENTIALEQUATIONS WITH CONSTANT COEFFICIENTSIn the last section we looked at a frictionless model of a vibrating spring from the perspectiveof a system of differential equations. We sketched solution curves in the position-velocityphase-plane, but we did[r]

Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. pp. 217–224. Springer, Heidelberg(1999)5. Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech Syst Signal Process. 5,81–88(1991). doi:10.1016/0888-3270(91)90016-X6. Glockle, WG, Nonnenmac[r]

2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8162.2 Integrating a System of Differential Equations by theMethod of Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8192.3 Finding Integrable Combinations . . . . . . . . . . . . . . . . . . .[r]

2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8162.2 Integrating a System of Differential Equations by theMethod of Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8192.3 Finding Integrable Combinations . . . . . . . . . . . . . . . . . . .[r]

the necessary condition (15.14.1.2) results in an equation with x, y,andz. Treating it asan algebraic (transcendental) equation for z,wefind z = z(x, y). The direct substitution ofthe expression z = z(x, y) into both equations (15.14.1.1) gives an answer to the questionwhether it is a solution of the[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 eine gesuchte[r]

mathematical statistics, etc. Special attention is paid to formulas (exact, asymptotical, andapproximate), functions, methods, equations, solutions, and transformations that are offrequent use in various areas of physics, mechanics, and engineering sciences.• The main distinction of this reference b[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 functions ϕp(x)andψq(y). Solving these systems, we get general[r]