Consider two-point boundary value problem {u\"(x) + q^2 u(x) - 1 = 0, x elemento

Question

Consider two-point boundary value problem {u"(x) + q^2 u(x) - 1 = 0, x elementof (a, b) a_11 u(a) + a_12 u'(a0 = g_l, a_12 u(b) + a_22 u'(b) = g_r, where > 0 and q are constants. Formally solve this problem and determine the required condition for u(x) to exist. This condition should be in terms of, q, a_ij, a, and b. Discuss also uniqueness/nonuniqueness of u(x). By appropriately choosing the values of a_ij, use the previous result to establish the existence of the problem when the various boundary condition types that we have studied in class are imposed. Here, you should discuss 10 combinations of boundary condition: D-D, D-N, D-R, N-D, N-N, N-R, R-D, R-N, R-R, and periodic. D stands for Dirichlet, N stands for Neumann, and R stands for Robin. The first letter in the combination is for condition on the left and the second letter is for condition on the right.

Explanation / Answer

let assume z(x) = q^2*u(x) -1

d2u/d2x = (1/q^2) d2z/d2x

So the differential equn will be

(-eps/q^2) * d2z/d2x + z = 0

d2z/d2x - k*z = 0 where k = q^2/eps > 0 as eps >0

auxilary equn will be

m^2 - k = 0

m = +/- sqrt(k) = +/- (p) where p = q/sqrt(eps)

soln

z = c1*exp(p*x) + c2*exp(-p*x)

q^2*u(x) -1 = c1*exp(px) -c2*exp(-px)

u(x) = 1/q^2[1 + c1*exp(px) -c2*exp(-px)] = k[1 + c1*y -c2/y]

u'(x) = p/q^2[c1*exp(px) + c2*exp(-px)] = k [c1*y -c2/y] y=exp(px)

put this two value in two boundary value problem. we will get two quadratic equation of y

A1*y(a)^2 + A2*y(a) + A3 = 0

B1*y(b)^2 + B2*y(b) + B3 = 0

From two equation we will get value of y(a) and y(b)

condition for existence of solution will be y(a) >0 and y(b) > 0 because exp(**) is always positive

using above conditions if we can eleminate c1 & c2 from the expression of u(x) then solution will be unique

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