Theory of Functionals and of Integral and by Vito Volterra

Theory of Functionals and of Integral and by Vito Volterra

By Vito Volterra

Книга conception of Functionals and of quintessential and Integro-Differential Equations conception of Functionals and of critical and Integro-Differential Equations Книги Математика Автор: Vito Volterra Год издания: 2005 Формат: pdf Издат.:Dover courses Страниц: 288 Размер: 14,3 Mb ISBN: 0486442845 Язык: Английский0 (голосов: zero) Оценка:A basic conception of the capabilities counting on a continuing set of values of one other functionality, this quantity is predicated at the author's basic idea of the transition from a finite variety of variables to a regularly endless quantity. bargains essentially with crucial equations, and in addition addresses the calculus of diversifications. 1930 version.

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Extra info for Theory of Functionals and of Integral and Integro-Differential Equations

Example text

Consider the \I ave equation It is easily seen that the functions u,,(x, y) = (x - y)", 11 = I, 2, ... satisfy the equation for each 11. ::; N. , for k = I, 2, ... i (2k - k=1 •)2k- l I)! \-;·-plane containing the origin. Hence. by the extended principle of superposition, the preceding series is also a solution of the wave equation. As a matter of fact, we notice that the series converges to the function u(x, y) = sin(x - y), which is easily seen to satisfy the wave equation. There is a variation of the principle of superposition concerning solution of a homogeneous differential equation that depends on a parameter.

When x 2 + 2y 2 = 4. + 2xy11 = 0, 11 = c" sin(x + I), when y 2 = 2x + 2 20. r11r 21. ) 11, xu,. 22. rn,. 23. Show that no solution exists for the differential equation of Problelll 22, which assullles the prescribed value ef>(_y) on the circle x 2 + y 2 = a 2 , unless cf> is of the 2 form qJ(x) = kc-x • where k is a constant. If cf> has the indicated form. show that the problem has infinitely many solutions. I.

1. In Problems 7 through 13, find the general solution of the given equation. 7. 9. + + Xllx Jllx Xllx - xy11,. - 2x11 x + 2 2 )'lly - xu = 2e" + x u, - xyu,. + 2y11 2 10. y. xu,. = 2 8. l. Jllr = = 2 x y. 0. 11. y ux - 12. (x 13. xyux - 14. Let Lu = Au, + Bur + Cu= + Du = 0 be a linear equation in three variables x, J', ::, where A, B, C, Dare constants. Introduce the new variables + y)(u, - 11r) + = 11 = y . «YJ. x 2 uy - yu = xy. ~ = a 1x + h1Y + I/ = a1 x + h2Y + ( = G3X + h3y + CI;: C2Z C3Z where the deterlllinant of the coefficients a;.

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