By L Dresner
This creation to the applying of Lie's idea to the answer of differential equations comprises labored examples and difficulties. The textual content exhibits how Lie's workforce conception of differential equations has purposes to either traditional and partial differential equations.
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Extra resources for Applications of Lie's Theory of Ordinary and Partial Differential Equations
Because the slope j(xo) is negative, a backward integration produces a positive solution on the interval (0. rOthe solution is negative and thus unacceptable. Hence, we have found solutions to Eq. 1) that are positive on the interval (0. ro. 2). Hence, to find them all, we have to integrate Eq. I ) numerically only once. Now the point P in the (p. r = 0 means moving along the separatrix S away from P in the direction of positive p . 2. The integral curve of Eq. xo) = 0, s ( . ~0/3. Because the integral curves converge in this direction, we expect the numerical integration to be stable.
3 . 1 ) must have the form where @ can be any function. From Eq. ( 3 . 3 . 4 ) it follows that there are real power-law solutions of the form y = A x @ when A is a real constant that satisfies the algebraic equation Under circumstances outlined below, one of them gives the asymptotic behavior of positive solutions on the half-line x 2 0 that vanish at infinity. The value of y at x = 0 specifies one of these solutions. (The determination of such a solution is called a two-point boundaryvalue problem.
All of these solutions are images of one another under the group ( 3 . 3 . 1 ) . Since any image of the point ( 0 , y l ( 0 ) ) is ( 0 . A P y l ( 0 ) ) , we can always find a value of A for which APy1 ( 0 ) = y z ( 0 ) . Furthermore, since the point x = oo transforms into the point x' = oo, we see that y 2 ( o o ) = A B y l ( o o ) = 0. Hence the solution y , ( x ) obeying the boundary conditions y 2 ( 0 ) = a2 and ~ ~ ( 0=0 0) is an image of the solution y , ( x ) obeying the boundary conditions y1(0) = a1 and y l ( o o ) = 0 , the value of A corresponding to the transformation being ( ~ ~ / a ~ ) ~ / ~ .
Applications of Lie's Theory of Ordinary and Partial Differential Equations by L Dresner