By Robert L. Borrelli
Книга Differential Equations: A Modeling point of view Differential Equations: A Modeling viewpoint Книги Математика Автор: Robert L. Borrelli, Courtney S. Coleman Год издания: 2004 Формат: djvu Издат.:Wiley Страниц: 736 Размер: 28,3 Mb ISBN: 0471433322 Язык: Английский0 (голосов: zero) Оценка:This powerful and functional re-creation maintains to target differential equations as a strong software in developing mathematical versions for the actual international. It emphasizes modeling and visualization of suggestions all through. each one bankruptcy introduces a version after which is going directly to examine ideas of the differential equations concerned utilizing an built-in analytical, numerical, and qualitative method. The authors current the fabric in a fashion that is transparent and comprehensible to scholars in any respect degrees. through the textual content the authors express their enthusiasm and pleasure for the research of ODEs.
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Additional info for Differential Equations: A Modeling Perspective
Our initial curve will be f(r) = (0, r), and as y = r along r, the initial condition is u(f(r)) = er. The equations for the (projected) characteristic curves are x~(s) =a, Xr(O) = 0, y~(s) = b, Yr(O) = r. The solution is Xr(s) =as, Yr(s) = r + bs. The ODEs along the characteristic curves are v~(s) = 0, Vr(O) =er. The solution is vr(s) = er. Now we need to express r, s in terms of x, y. First, s = a- 1 x, and next r = y-bs = y- (b/a)x. Thus, the solution of the PDE is u(x, y) = ey-(b/a)x = eY e-(b/a)x.
As before, we consider integral curves 'Y(r, s) ='Yr(s) = (x(r, s), y(r, s)) of V through an initial curve r = r(r). We find these solving the same ODEs as before, and the criterion for inverting the change of coordinates (r, s) t-t 3. s)) = VN(r, s) r. Substituting in (F1(x(r,s),y(r,s),~~~r,s), ... ,vN(r,s)))' FN(x(r, s), y(r, s), v1(r, s), ... ), and which is thus solvable at least for smalls. 6. Solve the system of ODE Ux + YUy = w, Wx + ywy = u, with initial conditions u(O, y) = y, w(O, y) = 1.
These examples also explain the role of boundary conditions. For instance, suppose n is a region in free space whose boundary is a perfect conductor. Rt) elsewhere (otherwise it would instantenously generate currents to arrange this). e. V is constant on the boundary. e. Vian is given), and one attempts to find the potential inside by solving Poisson's equation. 2. Euler-Lagrange equations Often PDE arise from somewhat different considerations, such as critical points of a functional, called an Euler-Lagrange functional.
Differential Equations: A Modeling Perspective by Robert L. Borrelli