applications of ordinary differential equations in civil engineering

... Wirtinger’s inequality For other study of Wirtinger’s inequality, one may see 15 and the references therein Now, we are ready to prove our main results We first give the proof of Theorem 1.3 Proof ... where ·, · is the inner product in Rp Then according to 14 , we give Wirtinger’s inequality and its proof T x Lemma 2.1 If x ∈ HT and 0, then t dt T x t T2 4π dt ≤ T x t ˙ dt 2.1 Proof By the assumptions, ... the proof is complete Corollary 2.3 Under the conditions of Lemma 2.1, the inequality inequality 2.1 Proof If x ∈ HT and T x t dt 2.4 0, then 2.1 follows 2.4 on taking z x implies Wirtinger’s...

Ngày tải lên: 22/06/2014, 02:20

... 710 Chapter 16 Integration of Ordinary Differential Equations CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood ... description of each of these types follows Runge-Kutta methods propagate a solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the ... routines; rkqs, bsstep, stiff, and stifbs are steppers; rkdumb and odeint are drivers Section 16.6 of this chapter treats the subject of stiff equations, relevant both to ordinary differential equations...

Ngày tải lên: 15/12/2013, 04:15

... step in a sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential ... discussion of the pitfalls in constructing a good Runge-Kutta code is given in [3] Here is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations You input ... Chapter 16 Integration of Ordinary Differential Equations yn yn + Figure 16.1.3 Fourth-order Runge-Kutta method In each step the derivative is evaluated four times: once at the initial point, twice...

Ngày tải lên: 15/12/2013, 04:15

... Ordinary Differential Equations } nrerror("Too many steps in routine odeint"); } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... free_vector(ak2,1,n); } Noting that the above routines are all in single precision, don’t be too greedy in specifying eps The punishment for excessive greediness is interesting and worthy of Gilbert and ... bound the error of each step individually In some applications you may be unusually sensitive about a “global” accumulation of errors, from beginning to end of the integration and in the worst possible...

Ngày tải lên: 24/12/2013, 12:16

... contains only even powers of h, 724 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential ... solution of a set of equations A second warning is that the techniques in this section are not particularly good for differential equations that have singular points inside the interval of integration ... in this section, is the best known way to obtain high-accuracy solutions to ordinary differential equations with minimal computational effort (A possible exception, infrequently encountered in...

Ngày tải lên: 24/12/2013, 12:16

... degree of robustness for problems with discontinuities Let us remind you once again that scaling of the variables is often crucial for successful integration of differential equations The scaling ... calculated during the integration The optimal column index q is then defined by 728 Chapter 16 Integration of Ordinary Differential Equations During the first step, when we have no information about ... "nrutil.h" #define KMAXX #define IMAXX (KMAXX+1) #define SAFE1 0.25 #define SAFE2 0.7 #define REDMAX 1.0e-5 #define REDMIN 0.7 #define TINY 1.0e-30 #define SCALMX 0.1 Maximum row number used in the extrapolation...

Ngày tải lên: 24/12/2013, 12:16

... Chapter 16 Integration of Ordinary Differential Equations Note that for compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations The values of y are ... rule for integrating y = f (x, y) for a system of n = nv/2 equations On input y[1 nv] contains y in its first n elements and y in its second n elements, all evaluated at xs d2y[1 nv] contains the ... possibility of a stiff set of equations arises Stiffness occurs in a problem where there are two or more very different scales of the independent variable on which the dependent variables are changing...

Ngày tải lên: 24/12/2013, 12:16

... those of Shampine simply by replacing the #define statements: #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define ... 16 #define #define #define #define #define #define C1X C2X C3X C4X A2X A3X Integration of Ordinary Differential Equations (1.0/2.0) (-3.0/2.0) (121.0/50.0) (29.0/250.0) 1.0 (3.0/5.0) indx=ivector(1,n); ... re-try step nrerror("exceeded MAXTRY in stiff"); 742 Chapter 16 #define #define #define #define #define C2X C3X C4X A2X A3X Integration of Ordinary Differential Equations -0.396296677520e-01 0.550778939579...

Ngày tải lên: 24/12/2013, 12:16

... The second of the equations in (16.7.9) is 752 Chapter 16 Integration of Ordinary Differential Equations you suspect that your problem is suitable for this treatment, we recommend use of a canned ... 750 Chapter 16 Integration of Ordinary Differential Equations For multivalue methods the basic data available to the integrator are the first few terms of the Taylor series expansion of the solution ... derivatives at a point, not the fact that we are solving a system of equations with many components y In terms of the data in (16.7.5), we can approximate the value of the solution y at some point x: y(x)...

Ngày tải lên: 24/12/2013, 12:16

... Medical Applications of Rapid Prototyping - A New Horizon Fig Segmentation using the software Fig Designing using CAD software Advanced Applications of Rapid Prototyping Technology in Modern Engineering ... Medical Applications of Rapid Prototyping - A New Horizon Fig Segmentation using the software Fig Designing using CAD software Advanced Applications of Rapid Prototyping Technology in Modern Engineering ... technologies in various engineering areas as the solutions to the real world engineering problems   Dr Md Enamul Hoque Associate Professor Department of Mechanical, Materials & Manufacturing Engineering...

Ngày tải lên: 27/06/2014, 11:20

... ODEs in Chemical Engineering • The ODEs that occur in Chemical Engineering tend to be of the following form: dx(t ) = f ( x, t ) dt • Following the previous steps, the ... x (t ) + k , t + h) School of Chemical Engineering and Advanced Materials Newcastle University 24 Solving sets of • st order ODEs Chemical engineering systems are often described by more than ... each integration interval, the equations are solved one at a time, using the same methods as for single ODEs, School of Chemical Engineering and Advanced Materials Newcastle University 25 Solving...

Ngày tải lên: 19/03/2015, 08:49

... Tissue Engineering Applications 33 3.1 Introduction As explained in chapter 1, inadequate vascularization is a major bottleneck limiting widespread clinical applications of tissue engineering, ... diabetic wounds 2.4 Potential applications of PHIs in tissue engineering As reviewed in chapter 1, clinical applications of engineered tissues are currently limited mainly to thin or avascular tissues ... number of obstacles preventing the realization of this goal 1.1.2 Vascularization is a major obstacle in tissue engineering While the field of tissue engineering has come a long way since its original...

Ngày tải lên: 09/09/2015, 11:11

... fundamental feature of the Poisson process that due to the independence of its increments the location of the set of points in the time interval [0, 1], see Fig 1.1.6, can be intuitively interpreted ... their increments over a period (s, t] are independent of As for t ≥ 0, s ∈ [0, t] 1.2 Supermartingales and Martingales Martingales As we will see later in the context of asset pricing and investing ... intensity appears to be large enough for realistic modeling of the dynamics of quantities in finance Here continuous trading noise and a few single events model the typical sources of uncertainty...

Ngày tải lên: 19/02/2014, 22:20

... Proof By the continuation of solutions and the singularity of ᏷ at the point P0 , the set ᏷(P0 ) = ∅ Taking into account the nature of the vector field and the definition of the singularity of ... ∂ω and the family ᐄ(P0 ) of all noncontinuable solutions of the initial value problem (2.6) By the continuity of the nonlinearity and the nature of the vector field (sign of f ), we have two possible ... a singular point of the above map ᏷ This is exactly the case, the existence of which we must investigate Lemma 2.5 [9] The above mapping ᏷ is upper semicontinuous (usc) at any nonsingular point...

Ngày tải lên: 22/06/2014, 22:20

... fundamental feature of the Poisson process that due to the independence of its increments the location of the set of points in the time interval [0, 1], see Fig 1.1.6, can be intuitively interpreted ... their increments over a period (s, t] are independent of As for t ≥ 0, s ∈ [0, t] 1.2 Supermartingales and Martingales Martingales As we will see later in the context of asset pricing and investing ... intensity appears to be large enough for realistic modeling of the dynamics of quantities in finance Here continuous trading noise and a few single events model the typical sources of uncertainty...

Ngày tải lên: 28/06/2014, 17:20

... phase lattice points can be arranged either in the form of grain inclusions, whiskers, 566 Applications of Monte Carlo Method in Science and Engineering fibers The influence of the input parameters ... exponent in the kinetic equations, particularly in small grain size 582 Applications of Monte Carlo Method in Science and Engineering regimes (i.e., for grain sizes ranging from to 15 lattice spacing ... second phase in the form of grains (5%) a), in the form of whiskers (5%) b), and in the form of fibers (10%) c) 2.3 Grain growth in two-phase materials When simulating grain growth in two-phase...

Ngày tải lên: 29/06/2014, 13:20

... and so forth In this book, Applications of Monte Carlo Method in Science and Engineering, we further expose the broad range of applications of Monte Carlo simulation in the fields of Quantum Physics, ... 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Applications of Monte Carlo Method in ... exited-state 10 Applications of Monte Carlo Method in Science and Engineering deactivation The energy level differences of some elements in a typical scintillator are in the range of some electronvolts...

Ngày tải lên: 29/06/2014, 13:20

... concept for civil engineering applications via developing self-healing concrete is examined in Chapter The selection of components of self-healing unit comprising of healing agent, container and ... self-healing polymer composites, tubular system is a more approachable method for applications in civil engineering area In civil engineering applications using mortar and concrete, because of the ... SELF-HEALING MATERIALS: NUMERICAL MODELLING AND APPLICATIONS IN CIVIL ENGINEERING TRAN DIEP PHUOC THAO (B.Eng Civil (Hons)) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL...

Ngày tải lên: 09/09/2015, 18:54

... presentation of methods as well as results in the area In recent years, there has been an increasing interest in extending certain classical results to differential equations in Banach spaces In this ... solutions of differential equations have been studied since the very beginning of this century The theory of almost periodic solutions has been developed in connection with problems of differential equations, ... abstract functional differential equations with infinite delay, there are three kinds of definitions of stabilities In Sections and 4, we prove some equivalence of these definitions of stabilities...

Ngày tải lên: 29/10/2015, 12:32

... broad-based disciplines such as industrial engineering Mechanical engineering has played, and continues playing, a commanding role in trends that drive change in engineering. ” As pointed out at a ... component of this engineering discipline KEY FINDINGS AND CONCLUSIONS The key findings and conclusions of the report are summarized below The United States is Among the Leaders in Mechanical Engineering ... president of the National Academy of Engineering The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions...

Ngày tải lên: 06/03/2014, 15:20