examples of linear and nonlinear differential equations

... EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND DOCUMENTATION OF SAP2000. THE PROGRAM HAS BEEN THOROUGHLY TESTED AND USED. IN USING THE PROGRAM, HOWEVER, THE USER ACCEPTS AND UNDERSTANDS ... 1 - 2 Overview of the Program 1 Overview of the Program SAP2000 is a stand-alone finite-element-based structural program for the analysis and design of civil structures. It offers an intuitive, ... spans 60 feet, and has a width and height of 12 feet each. The supports are roll- ers at one end, and pins at the other. The trusses and cross members are to be constructed of 2L4X4’s, while...

Ngày tải lên: 06/09/2012, 15:56

... Dimensions and tolerances – merge, selection, and snap tolerances; font sizes; zoom increment; and others  Design codes and their parameters  Colors of objects and results for display and printing ... i.e., as data organized in a set of tables with specified table names and column headings. These data include the definition of the model and the results of analysis and design. Tabular data can ... objects and elements. Groups A group is a named collection of objects. It may contain any number of objects of any number of types. Groups have many uses, including:  Quick selection of objects...

Ngày tải lên: 01/04/2014, 03:20

... Recently, there are a large number of papers dealing with the existence of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis (fixed point theore ... system of nonlinear fractional differential equations. Int J Differ Equ 2010, 2010:1-12. 28. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations ... existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations. 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and...

Ngày tải lên: 21/06/2014, 03:20

... biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. 1; for an overview of existing results and of recent research areas of ... problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986. 41 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations, ” Journal of Mathematical ... hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic...

Ngày tải lên: 21/06/2014, 16:20

... Wang, and L. Ge, “Existence of nonoscillatory solution of high-order nonlinear difference equation,” Nonlinear Dynamics and Systems Theory, vol. 6, no. 2, pp. 205–210, 2006. 33 E. M. E. Zayed and ... behaviour of solutions of second- and third-order nonlinear differential equations in recent years. See, for instance, 1–37 as some related papers or Journal of Inequalities and Applications 7 Proof. ... J C. Liang, and D H. Wang, “Existence of nonoscillatory solution of third order linear neutral delay difference equation with positive and negative coefficients,” Nonlinear Dynamics and Systems...

Ngày tải lên: 22/06/2014, 02:20

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Ngày tải lên: 06/08/2014, 05:20

... analysis (FA) and partial differential equations (PDEs). The first part deals with abstract results in FA and operator theory. The second part concerns the study of spaces of functions (of one or ... They appear in linear and nonlinear PDEs that arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, and physics. They belong to the toolbox of any graduate ... {x 0 } and C. Proof of Lemma 1.3. After a translation we may always assume that 0 ∈ C.We may thus introduce the gauge p of C (see Lemma 1.2). Consider the linear subspace G = Rx 0 and the linear...

Ngày tải lên: 04/02/2014, 11:10

... the general solution of the differential equation. Chapter 0 Ordinary Differential Equations 3 Principle of Superposition. If u 1 (t) and u 2 (t) are solutions of the same linear homogeneous equation ... acknowledge the skillful work of Cindy Smith, who was the LaTeX compositor and corrected many of my mistakes, the help of Academic Press editors and consultants, and the guidance of reviewers for this ... follow the derivations of the heat and wave equations. The principal objective of the book is solving boundary value problems involving partial differential equations. Separation of variables receives...

Ngày tải lên: 17/02/2014, 14:20

... connected subset Σ of R n (Σ is the state space and elements of Σ are called states) and (b) two C 2 functions  T :Σ→ (0, ∞) Λ:Σ→ R n . T is the temperature and the components of Λ are the generalized ... equilibrium B. Examples 1. Simple fluids 2. Other examples C. Physical interpretations of the model 1. Equilibrium 2. Positivity of temperature 3. Extensive and intensive parameters 4. Concavity of S 5. ... Entropy and irreversibility A. A model material 1. Definitions 2. Energy and entropy a. Working and heating b. First Law, existence of E c. Carnot cycles d. Second Law e. Existence of S 3. Efficiency of...

Ngày tải lên: 17/03/2014, 14:29

Ngày tải lên: 31/03/2014, 15:16

... an offshoot of the investigation of the corresponding questions for nonlinear waves. The nonlinear problems required a more powerful approach altogether, and eventually the possibility of using ... in pressure, density, and velocity: the blast waves of explosions and the sonic booms of high speed aircraft. But the whole intricate machinery of nonlinear hyperbolic equations had to be developed ... in gas dynamics led to most of the fundamental ideas in nonlinear hyperbolic waves. The most outstanding new phenomenon of the nonlinear theory is the appearance of shock waves, which are abrupt...

Ngày tải lên: 12/06/2014, 16:26

... Lower and Upper Solutions Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs. Since the task of finding a pair of lower and ... of α and β and r ≤ x ≤ 1/r. Therefore Theorem 5.4 implies the existence of a weak solution of 7.7 between α and β. Moreover, since f is positive between α and β the solution is increasing and, ... ε, t 0  L, and since ε ∈ 0,L was fixed arbitrarily, the proof of Step 4 is complete. Conclusion The construction of y ∞ and Step 1 imply that y ∞ ≥ x ∗ and the definition of x ∗ and Step 4 imply...

Ngày tải lên: 21/06/2014, 20:20

... 5.6 gives a characterization of a class of nonlinear periodic problems 5.1a and 5.1b which have only one solution. We begin the investigation of problem 5.1a and 5.1b with a uniqueness result. Theorem ... discontinuous for certain values of the time variable t ∈ 0,T.Thisformoff is motivated by a variety of initial and boundary value problems known from applications and having nonlinear, discontinuous forcing ... in both, the case of intersecting solutions u 1 and u 2 and the case of separated u 1 and u 2 , there exists an interval α, β ⊂ 0,T satisfying 5.5. Now, by 5.1a, 5.3 ,and 5.5,weobtain v   t  >  a t −...

Ngày tải lên: 21/06/2014, 20:20

... conditions of Theorem 1.6. 4. Examples From the main results of the present paper new and optimal in some sense sufficient condi- tions for the existence of periodic solutions of linear and sublinear ... one and only one ω-periodic solution. 10 Boundary Value Problems 3. Proof of the main results Proof of Theorem 1.1. Without loss of generality, it can be assumed that h00. On the other hand, ... Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol.56ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1977. 5 L. A. Kipnis, “On periodic solution of...

Ngày tải lên: 21/06/2014, 22:20

... prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results of rational difference equations. For the systematical ... Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. 2 R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, vol. 155 of Mono- graphs and ... and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992. 3 V. L. Koci ´ c a nd G. Ladas, Global Behavior of Nonlinear D i fference Equations of Higher Order with...

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Ngày tải lên: 23/06/2014, 00:20