Verify (a) and (d). pr solution. IV. Fundamental Solutions. Let J = [a,b], let Q be the

Verify (a) and (d).

 

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pr solution. IV. Fundamental Solutions. Let J = [a,b], let Q be the square J x J in the re-plane, and let Q₁ be the triangle a ≤ ≤x≤ b, Q2 be the triangle a ≤ x ≤ { ≤b. belongs to Note that both triangles are closed and that the diagonal x = both triangles. A function y(x, E) defined in Q is called a fundamental solution of the homogeneous differential equation (2) Lu= 0 if it has the following properties (recall that p > 0): (a) y(x, ) is continuous in Q. (b) The partial derivatives Y Yra exist and are continuous in Q₁ and Q2 (on the diagonal one has to take the one-sided derivatives from the corresponding omogeneous triangle). (c) Let & E J be fixed. Then y(x, E), considered as a function of æ, is a solution to Ly = 0 for x , x EJ. the first derivative makes a jump of magnitude (d) On the diagonal x = 1/p; i.e., 1 p(x) 1 Yx (x+,x) - Yx (x,x) Here, Yz(x+,x) is the right-sided derivative of y(x, E) with respect to x at the point (x,x) (or, equivalently, the limit of 7 when the point (x,x) is approached from the right); the left-sided derivative z(x-,x) is defined similarly. for a < x < b. pr solution. IV. Fundamental Solutions. Let J = [a,b], let Q be the square J x J in the re-plane, and let Q₁ be the triangle a ≤ ≤x≤ b, Q2 be the triangle a ≤ x ≤ { ≤b. belongs to Note that both triangles are closed and that the diagonal x = both triangles. A function y(x, E) defined in Q is called a fundamental solution of the homogeneous differential equation (2) Lu= 0 if it has the following properties (recall that p > 0): (a) y(x, ) is continuous in Q. (b) The partial derivatives Y Yra exist and are continuous in Q₁ and Q2 (on the diagonal one has to take the one-sided derivatives from the corresponding omogeneous triangle). (c) Let & E J be fixed. Then y(x, E), considered as a function of æ, is a solution to Ly = 0 for x , x EJ. the first derivative makes a jump of magnitude (d) On the diagonal x = 1/p; i.e., 1 p(x) 1 Yx (x+,x) - Yx (x,x) Here, Yz(x+,x) is the right-sided derivative of y(x, E) with respect to x at the point (x,x) (or, equivalently, the limit of 7 when the point (x,x) is approached from the right); the left-sided derivative z(x-,x) is defined similarly. for a < x < b.

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A A fundamental solution of the homogeneous differential equation 2 Lu 0 is a function gammax xi defined in Q that satisfies the following properties a gammax xi is continuous in Qb gammax xi has cont... View the full answer