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Download Numerical Methods for Grid Equations: Volume I Direct by Aleksandr A. Samarskii, Evgenii S. Nikolaev (auth.) PDF

By Aleksandr A. Samarskii, Evgenii S. Nikolaev (auth.)

The finite-difference answer of mathematical-physics differential equations is performed in phases: 1) the writing of the adaptation scheme (a range­ ence approximation to the differential equation on a grid), 2) the pc answer of the adaptation equations, that are written within the type of a excessive­ order procedure of linear algebraic equations of exact shape (ill-conditioned, band-structured). software of common linear algebra tools isn't continuously applicable for such platforms as a result of have to shop a wide quantity of knowledge, in addition to end result of the great amount of labor required by means of those tools. For the answer of distinction equations, detailed tools were built which, in a single method or one other, consider designated beneficial properties of the matter, and which enable the answer to be chanced on utilizing much less paintings than through the final tools. This paintings is an extension of the e-book distinction M ethod3 for the answer of Elliptic Equation3 by way of A. A. Samarskii and V. B. Andreev which thought of a complete set of questions hooked up with distinction approximations, the con­ struction of distinction operators, and estimation of the ~onvergence price of distinction schemes for normal elliptic boundary-value difficulties. the following we give some thought to in basic terms answer tools for distinction equations. The ebook in truth includes volumes.

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Additional resources for Numerical Methods for Grid Equations: Volume I Direct Methods

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A2(i + 1) a2(O) [y(I) - yeO)]. Taking into account the initial values for v2(i), we obtain (26) Summing the left- and right-hand sides of (26) for i between 0 and k - 1, we get k-l v2(k) = V2(O) + 3 LW - i + 1) = k(k 2 - 3k + 5). ;=0 Thus, the particular solutions of the homogeneous equation (25) are vl(k) == 1, (27) and the general solution (25) has the form We now construct a particular solution to the non-homogeneous equation (23). Substituting (24) and (27) into the formula (19), we obtain (28) Here (26) was used.

V m ) == o. Consider the system of equations + C2 V 2(io) + ... + + 1) + C2V2(io + 1) + ... + cmvm(io)= O. cmvm(io + 1)= O. CIVI(iO) CIVI(iO (5) Since the determinant a i ( vI, ... ,vm ) of this system is, by assumption, equal to zero, there exists a non-zero solution CI, C2, ... ,Cm to this system. Consequently, for these CI, C2, ... , Cm, equation (4) is valid for i = i o, io + 1, ... , io + m - 1. We now show that (4) is valid for i = io + m. For this, we take equation (1) with 1= 1,2, ...

The lemma is proved. 2 Theorems about the solutions of linear equations. First we will prove a theorem about the general solution of the homogeneous linear equation (1). Theorem 2. If VI (i), v2(i), ... , vm(i) are linearly independent solutions of equation (1), then the general solution of this equation has the form (6) where CI, C2, ... ,Cm are arbitrary constants. Proof. In fact, by theorem 1 the function y(i) defined by formula (6) is a solution to equation (1). We will now show that all solutions of equation (1) are of this form.

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