using power series to solve ordinary differential equations
an introduction to ordinary differential equations
... INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS This refreshing, introductory textbook covers standard techniques for solving ordinary differential equations, as well as introducing students to qualitative ... will consider only ordinary differential equations It is possible to write these two equations much more concisely using vector calculus notation Imagine that ∇ represents a vector of partial derivatives, ... systematic way 3.1 Ordinary and partial differential equations The most significant distinction is between ordinary and partial differential equations, and this depends on whether ordinary or partial...
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Báo cáo hóa học: " Research Article A Generalized Wirtinger’s Inequality with Applications to a Class of Ordinary Differential Equations" docx
... 1.1 and the nonautonomous equation 1.2 We first give a generalized Wirtinger’s inequality Then we turn to consider the problems on 1.1 and 1.2 by using the inequality In order to state our main ... delay equations with three delays via bi-Hamiltonian ¸ systems,” Nonlinear Analysis: Theory, Methods & Applications, vol 64, no 11, pp 2433–2441, 2006 11 S Jekel and C Johnston, “A Hamiltonian ... Society, vol 238, pp 139–164, 1978 J L Kaplan and J A Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, ” Journal of Mathematical Analysis and Applications,...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... 16.6 of this chapter treats the subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL ... than Runge-Kutta In recent years Bulirsch-Stoer has been replacing predictor-corrector in many applications, but it is too soon to say that predictor-corrector is dominated in all cases However, ... sophisticated predictor-corrector routines are competitive Accordingly, we have chosen not to give an implementation of predictor-corrector in this book We discuss predictor-corrector further in §16.7,...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
... yout[i]=y[i]+h6*(dydx[i]+dyt[i]+2.0*dym[i]); weights free_vector(yt,1,n); free_vector(dyt,1,n); free_vector(dym,1,n); 714 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER ... along the trajectory of an ordinary differential equation can serve as an initial point The fact that all steps are treated identically also makes it easy to incorporate Runge-Kutta into relatively ... in mind, however, that the old workhorse’s last trip may well be to take you to the poorhouse: Bulirsch-Stoer or predictor-corrector methods can be very much more efficient for problems where very...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... errmax,h,htemp,xnew,*yerr,*ytemp; 720 Chapter 16 Integration of Ordinary Differential Equations ak2=vector(1,n); ak3=vector(1,n); ak4=vector(1,n); ak5=vector(1,n); ak6=vector(1,n); ytemp=vector(1,n); for (i=1;i
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... 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 Equations ... midpoint method to the Bulirsch-Stoer technique (§16.4) derives from a “deep” result about equations (16.3.2), due to Gragg It turns out that the error of (16.3.2), expressed as a power series in h, ... caveats, we believe that the Bulirsch-Stoer method, discussed in this section, is the best known way to obtain high-accuracy solutions to ordinary differential equations with minimal computational...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is best to reformulate them as a system of first-order equations, as discussed ... carried out using quantities 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 729 d=matrix(1,nv,1,KMAXX); err=vector(1,KMAXX); x=vector(1,KMAXX); yerr=vector(1,nv); ysav=vector(1,nv); ... communication theory to determine the “average expected convergence behavior” of the extrapolation His model produces certain correction factors α(k, q) by which Hk is to be multiplied to try to get convergence...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
... With this storage arrangement you can use bsstep simply by replacing the call to mmid with one to stoerm using the same arguments; just be sure that the argument nv of bsstep is set to 2n You ... not referenced Also input is htot, the total step to be taken, and nstep, the number of substeps to be used The output is returned as yout[1 nv], with the same storage arrangement as y derivs ... 16.5 Second-Order Conservative Equations 733 Here zm is y (x0 + H) Henrici showed how to rewrite equations (16.5.2) to reduce roundoff error by using the quantities ∆k ≡ yk+1 − yk Start...
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
... free_vector(ysav,1,n); free_vector(g4,1,n); free_vector(g3,1,n); free_vector(g2,1,n); free_vector(g1,1,n); free_vector(err,1,n); free_vector(dysav,1,n); free_matrix(dfdy,1,n,1,n); free_vector(dfdx,1,n); free_matrix(a,1,n,1,n); ... Multivalue, and Predictor-Corrector Methods 747 free_vector(ysav,1,nv); free_vector(yerr,1,nv); free_vector(x,1,KMAXX); free_vector(err,1,KMAXX); free_matrix(dfdy,1,nv,1,nv); free_vector(dfdx,1,nv); ... dfdx=vector(1,n); dfdy=matrix(1,n,1,n); dysav=vector(1,n); err=vector(1,n); g1=vector(1,n); g2=vector(1,n); g3=vector(1,n); g4=vector(1,n); ysav=vector(1,n); xsav=(*x); Save initial values for (i=1;i
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
... are solved the same way as implicit multistep methods: either by a predictor-corrector approach using an explicit method for the predictor, or by Newton iteration for stiff systems Why go to all ... 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 ... that can be used to control accuracy and to adjust stepsize If one corrector step is good, aren’t many better? Why not use each corrector as an improved predictor and iterate to convergence on...
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Báo cáo khoa học: "Using comparable corpora to solve problems difficult for human translators" pptx
... implemented in ASSIST, a tool that helps translators to nd solutions for difcult translation problems The tool presents the results as lists of translation suggestions (usually 50 to 100 items) ordered ... questionnaires to the evaluators The segments were presented in sentence context and translators had an option of providing their own solutions and comments Table shows one of the questions sent to evaluators ... BNC) in comparison to ữồũờ ùợóỡỡ in Russian (70) Other translation equivalents in Table are generated by ASSIST We then asked professional translators afliated to a translators association (identity...
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ordinary differential equations and dynamical systems - g. teschl
... Introduction §1.1 Newton’s equations §1.2 Classification of differential equations §1.3 First order autonomous equations §1.4 Finding explicit solutions 11 §1.5 Qualitative analysis of first order equations ... Transform both equations into a first order system (ii) Compute the solution to the approximate system corresponding to the given initial condition Compute the time it takes for the stone to hit the ... computer program like Mathematica to solve differential equations for us For example, to solve x = sin(t)x ˙ (1.57) you would use the command In[1]:= Out[1]= DSolve[x [t] == x[t]Sin[t], x[t], t]...
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Báo cáo toán học: " Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments" pptx
... of solutions to first-order ordinary differential equations with deviated arguments Delay differential equations are included in our general framework, which even allows deviations to depend on ... γ(τ (s, γ))))ds t0 It is an elementary matter to check that T is a completely continuous operator from C(I) into itself (one has to take Remark into account) Therefore, Schauder’s Theorem ensures ... problems for ordinary differential equations with deviated arguments J Optim Theory Appl 135(2), 257–269 (2007) Jankowski, T: Existence of solutions of boundary value problems for differential equations...
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Báo cáo hóa học: " Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space" docx
... almost automorphic mild solutions to some semilinear abstract differential equations Semigroup Forum 2004, 69:80-86 14 Zhao Z-H, Chang Y-K, Nieto JJ: Almost automorphic and pseudo almost automorphic ... to some stochastic hyperbolic differential equations Electron J Differential Equations 2009, 111:1-14 19 Da Prato G, Tudor C: Periodic and almost periodic solutions for semilinear stochastic ... neutral functional differential equations Comput Math Appl 2008, 55:2593-2601 Diagana T, N’Guérékata GM: Almost automorphic solutions to semilinear evolution equations Funct Differential Equation...
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Báo cáo hóa học: "TERMINAL VALUE PROBLEM FOR SINGULAR ORDINARY DIFFERENTIAL EQUATIONS: THEORETICAL ANALYSIS AND NUMERICAL SIMULATIONS OF GROUND STATES" pdf
... BVP [13] W Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, vol 182, Springer, New York, 1998 [14] W Wasov, Asymptotic Expressions for Ordinary Differential Equations, John ... and free boundary problems for quasilinear elliptic operators, Advances in Differential Equations (2000), no 1–3, 1–30 [8] K Kuratowski, Topology II, Academic Press, New York, 1968 [9] P K Palamides, ... same time, to the methods of [9, 11] or [10] For the convenience of the reader and to make the paper self-contained, we summarize here the basic notions used in the sequel First, we refer to the...
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Numerical Methods for Ordinary Differential Equations Butcher Tableau doc
... correspond to incomplete eigenvector spaces and the need to use generalized eigenvectors Also, in this special case, the eigenvector corresponding to λ, together with the generalized eigenvectors if ... Taylor Series Methods 250 Introduction to Taylor series methods 251 Manipulation of power series 252 An example of a Taylor series solution 253 Other methods using higher ... and Difference Equations 10 Differential Equation Problems 100 Introduction to differential equations As essential tools in scientific modelling, differential equations are familiar to every educated...
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Báo cáo nghiên cứu khoa học: "USING ADAPTIVE CONTROL TO SOLVE THE TRACKING PROBLEM FOR A MOBILE MANIPULATOR" pptx
... arc length of torch is proposed The simulations using MatLab V6.5 and Simulink V5.1 are also performed to show the effectiveness of the proposed controller The paper also shows how to get the tracking ... Manipulator The following constraints will be examined for choosing the configuration of the mobile manipulator The orientation of the torch should lie on the tangent plane of the welding trajectory ... in welding process According to the above conditions, in the configuration of the manipulator, the torch orientation is fixed on the tilt of 45 degrees with respect to the link direction of the...
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NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
... Integration via Taylor Series Expansion • Alternative methods have to be employed and all are based on the Taylor Series • Note: any function can be expanded as a Taylor Series x(t ) = x(t0 ) ... + (t − t0 ) x (t0 ) + (t − t0 ) x (4)' (t0 ) +K 3! 4! ' • Taylor Series expansion of x(t) about t0 The series can be used to solve School of Chemical Engineering and Advanced Materials Newcastle ... Solution using Taylor Series x (t ) = x (t ) + ( t − t ) x (t ) + ( t − t ) x '' (t ) + ' • • Accuracy of this solution is dependent on the ‘order’ of the Taylor Series expansion Need to have...
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Tài liệu AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS ppt
... blank AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, ... the term characteristics to denote them for short There are several ways to compute the characteristics One of them is to solve the full characteristic equations, and then to project the solution ... attempting to solve specific problems that arise in applications Therefore we took great care to create a balanced exposition of the theoretical and applied facets of PDEs The book is flexible enough to...
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