solution of partial differential equations
Tài liệu Numerical Solution of Stochastic Differential Equations with Jumps in Finance pdf
Ngày tải lên :
19/02/2014, 22:20
... approximation of continuous solutions of SDEs The discrete time approximation of SDEs with jumps represents the focus of the monograph The reader learns about powerful numerical methods for the solution of ... Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical ... f first derivative of f : f second derivative of f : f : Q1 → Q function f from Q1 into Q2 ∂u ∂xi ith partial derivative of u : ∂ k ∂xi u → → d → kth order partial derivative of u with respect...
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Numerical Solution of Stochastic Differential Equations with Jumps in Finance docx
Ngày tải lên :
28/06/2014, 17:20
... approximation of continuous solutions of SDEs The discrete time approximation of SDEs with jumps represents the focus of the monograph The reader learns about powerful numerical methods for the solution of ... Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical ... f first derivative of f : f second derivative of f : f : Q1 → Q function f from Q1 into Q2 ∂u ∂xi ith partial derivative of u : ∂ k ∂xi u → → d → kth order partial derivative of u with respect...
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The numerical solution of delay-differential equations P1
Ngày tải lên :
11/10/2014, 04:02
... Delay -differential equations In this thesis we are concerned with the numerical solution of delay -differential equations (DDE's) Delay -differential equations may best be regarded as extensions of ... the propagation of derivative discontinuities through systems of equations is presented Following a short resumd of standard techniques for the solution of ordinary differential equations (ODE's), ... for the solution of systems of delay -differential equations A novel representation for the differential equation is used to acknowledge structural differences between delay- and ordinary differential...
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The numerical solution of delay-differential equations P2
Ngày tải lên :
11/10/2014, 04:03
... of the root location algorithm discussed in (2.7) Bibliography 415 418 291 appendix A a formal proof for (1.1.3:14) 292 A formal proof for (1.1.3:14) In this appendix we give a formal proof of ... transitive closure of E Heuristically, by an extension of the above arguments, we may think of Ei as defining the set of points whose continuity bounds can affect the continuity bound, ci , of yi El ... the notation of (5), s q Es q+ whereas t q ,Et 1+ Thus k' < k cannot be true and the result then follows [] We can now use this result to prove part (ii) of Proposition 1: PROOF of Proposition...
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Báo cáo y học: "Modelling of oedemous limbs and venous ulcers using partial differential equations" pdf
Ngày tải lên :
13/08/2014, 23:20
... measurement of the amount of oedema as well as the area and volume of the ulcer wounds This is because without an accurate and objective means of measuring changes in the size or shape of ulcers, ... in the linearity of the reflected stripe) [8] or stereo-photogrammetry The availability of high-resolution 3D digital cameras, increasing computing power and the development of software techniques ... exponential functions The specific forms of A0 (u), An (u), Bn (u) and R(u, v) for the case of Equations (1) can be found in [17] and for the case of Equations (2) can be found in [21] The main...
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Partial Differential Equations part 1
Ngày tải lên :
28/10/2013, 22:15
... What are the variables? • What equations are satisfied in the interior of the region of interest? • What equations are satisfied by points on the boundary of the region of interest? (Here Dirichlet ... conceptually, to the solution of large numbers of simultaneous algebraic equations When such equations are nonlinear, they are usually solved by linearization and iteration; so without much loss of generality ... problem as being the solution of special, large linear sets of equations As an example, one which we will refer to in §§19.4–19.6 as our “model problem,” let us consider the solution of equation (19.0.3)...
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Partial Differential Equations part 2
Ngày tải lên :
07/11/2013, 19:15
... coefficients of the difference equations are so slowly varying as to be considered constant in space and time In that case, the independent solutions, or eigenmodes, of the difference equations are all of ... j 838 Chapter 19 Partial Differential Equations stable unstable ∆t ∆t ∆x ∆x x or j (a) ( b) Figure 19.1.3 Courant condition for stability of a differencing scheme The solution of a hyperbolic ... rewritten as 840 Chapter 19 Partial Differential Equations Other Varieties of Error ξ = e−ik∆x + i − v∆t ∆x sin k∆x (19.1.25) An arbitrary initial wave packet is a superposition of modes with different...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
Ngày tải lên :
15/12/2013, 04:15
... subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley) Lapidus, L., and Seinfeld, J 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press) ... information about the quality of the solution after the step The routine is dumb, however, and it is unable to make any adaptive decision about whether the solution is of acceptable quality or not...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... 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 ... is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations You input the values of the independent variables, and you get out new values which are stepped ... free_vector(dym,1,n); 714 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A 1964, Handbook of Mathematical Functions, Applied...
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Tài liệu Partial Differential Equations part 3 pptx
Ngày tải lên :
15/12/2013, 04:15
... 19 Partial Differential Equations The physical interpretation of the restriction (19.2.6) is that the maximum allowed timestep is, up to a numerical factor, the diffusion time across a cell of ... to evolve through of order λ2 /(∆x)2 steps before things start to happen on the scale of interest This number of steps is usually prohibitive We must therefore find a stable way of taking timesteps ... amplitudes, so that the evolution of the larger-scale features of interest takes place superposed with a kind of “frozen in” (though fluctuating) background of small-scale stuff This answer gives...
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Tài liệu Partial Differential Equations part 4 ppt
Ngày tải lên :
15/12/2013, 04:15
... PDEs When you increase the order of a differencing method to greater than the order of the original PDEs, you introduce spurious solutions to the difference equations This does not create a problem ... αx sin ky ∆)2 856 Chapter 19 Partial Differential Equations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator splitting or time ... (19.3.16) The advantage of this method is that each substep requires only the solution of a simple tridiagonal system Operator Splitting Methods Generally The basic idea of operator splitting,...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
Ngày tải lên :
15/12/2013, 04:15
... effectively fewer equations than unknowns In this case there can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution ... combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding the solution space of A involves • Singular value decomposition of a matrix ... written as the N ×N set of equations (AT · A) · x = (AT · b) (2.0.4) where AT denotes the transpose of the matrix A Equations (2.0.4) are called the normal equations of the linear least-squares...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
Ngày tải lên :
15/12/2013, 04:15
... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... the identity matrix, of course) • Interchanging any two columns of A gives the same solution set only if we simultaneously interchange corresponding rows of the x’s and of Y In other words, this ... column we are about to eliminate Partial pivoting is easier than full pivoting, because we don’t have to keep track of the permutation of the solution vector Partial pivoting makes available...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
Ngày tải lên :
15/12/2013, 04:15
... Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to solve the next set of ... float **qt, int n, int i, float a, float b); int i,j,k; 102 Chapter Solution of Linear Algebraic Equations We will make use of QR decomposition, and its updating, in §9.7 CITED REFERENCES AND ... the second column below the second element, and so on up to Qn−1 Thus 100 Chapter Solution of Linear Algebraic Equations for (j=1;j
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Tài liệu Partial Differential Equations part 5 ppt
Ngày tải lên :
15/12/2013, 04:15
... level of CR, we have reduced the number of equations by a factor of two Since the resulting equations are of the same form as the original equation, we can repeat the process Taking the number of ... North America) ∂u = g(y) ∂x 862 Chapter 19 Partial Differential Equations The finite-difference form of equation (19.4.28) can be written as a set of vector equations uj−1 + T · uj + uj+1 = gj ∆2 ... above solution a solution uH of the homogeneous equation ∂2u ∂2u + =0 ∂x2 ∂y (19.4.11) that satisfies the required boundary conditions In the continuum case, this would be an expression of the...
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Tài liệu Partial Differential Equations part 6 doc
Ngày tải lên :
15/12/2013, 04:15
... ease of programming outweighs expense of computer time Occasionally, the sparse matrix methods of §2.7 are useful for solving a set of difference equations directly For production solution of large ... Chapter 19 Partial Differential Equations ADI (Alternating-Direction Implicit) Method The ADI method of §19.3 for diffusion equations can be turned into a relaxation method for elliptic equations ... application of the above results, consider our model problem for which ρJacobi is given by equation (19.5.11) Then equations (19.5.19) and (19.5.20) give 868 Chapter 19 Partial Differential Equations...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
Ngày tải lên :
24/12/2013, 12:16
... generally useful stepper routine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step Call that y[1 n] Let us require ... steps (see Figure 16.2.1) How much overhead is this, say in terms of the number of evaluations of the right-hand sides? Each of the three separate Runge-Kutta steps in the procedure requires ... also occur to you that, ignoring terms of order h6 and higher, we can solve the two equations in (16.2.1) to improve our numerical estimate of the true solution y(x + 2h), namely, y(x + 2h) =...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
Ngày tải lên :
24/12/2013, 12:16
... 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 ... method does an excellent job of feeling its way through rocky or discontinuous terrain It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations A second warning ... techniques in this section are not particularly good for differential equations that have singular points inside the interval of integration A regular solution must tiptoe very carefully across such points...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
Ngày tải lên :
24/12/2013, 12:16
... method a 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 ... 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 ... is a particular class of equations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly The equations are second-order...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
Ngày tải lên :
24/12/2013, 12:16
... 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 stored ... vol 27, pp 505–535 16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set of equations arises Stiffness occurs in ... 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 −...
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