Verify (a) and (d).

 

pr solution. IV. Fundamental Solutions. Let J = [a,b], let Q be the square J x J in the re-plane, and let Q 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 Q 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 , 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.