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only 5 questions on file (Assignment 2.pdf )

Detailed explanation

Math 7731 – Mathematical Problems in Industry

Assignment 2 (2017)

1. In lectures we defined the Sobolev space ) with the norm

.

However, since ), it will also have the norm associated with the Sobolev space H1(V ),

.

We want to show that these are equivalent norms. That is, there are constants k,K 0 such that for any

),

. (1)

(a) Show that if (1) is true, then there also exists k0,K0 0 such that

. (2)

So equivalence is symmetric, as we would expect.

(b) We will show this equivalence for the one-dimensional case where V is the interval (a,b), though it is true for n = 2,3 dimensions also. The left hand inequality of (1) is obvious (with k = 1). Let f be any bounded and continuous function with bounded and continuous first derivatives, and which is zero on the boundary of V , that is at x = a and x = b. Then for any x ? (a,b),

where f0 = df/dx. Why?

(c) Apply the Cauchy-Schwarz inequality to the right hand side of this to obtain

.

(d) Show that

|f(x)|2 = ||f0||2L2|x - a| where ||.||L2 is the L2 norm defined in lectures.

(e) By integrating both sides with respect to x over the entire interval (a,b), conclude that

||f||2L2 = C||f0||2L2

for some C 0 which does not depend on f. What does C depend upon?

(f) Use the result (e) to prove the right hand inequality of (1).

(g) We have now proved (1) for the special case of f as described in (b). How do you think we would prove it for any )? (Describe in words.)

2. We want to prove the result that there exists M 0 such that for any f ? H1(V ),

. (3)

Intuitively, this result is saying that the size of f on V is “controlled” by the size of f on the boundary and the size of the first partial derivatives of f on V , where size is measured by the L2 norm. Again we will only consider the case where V is the interval (a,b), though the result is true for n = 2,3 dimensions also. In the one-dimensional case the boundary integral over ?V on the right hand side takes the simple form f(a)2 + f(b)2.

(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 1(b), show that or any x ? V ,

,

and so

(c) Integrate both sides with respect to x over the entire interval (a,b) to conclude that

where C does not depend on f. What does C depend upon?

(d) Deduce from (3) that there exists M0 0 such that for any f ? H1(V ),

. (4)

3. We next want to prove the result that there exists M 0 such that for any f ? H1(V ), the boundary integral

Z

f2 dS = M||f||2H1. (5)

?V

Intuitively, this is saying that the size of f on the boundary is controlled by the H1 norm of f on V . Again we will only consider the case where V is the interval (a,b), though the result is true for n = 2,3 dimensions also. In this one-dimensional case the boundary integral on the left hand side takes the simple form f(a)2 + f(b)2.

(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 2(a), show that or any x ? V ,

,

and

(c) Integrate both sides with respect to x over the entire interval (a,b) to conclude that

|f(a)|2 = C||f||2H1

where C does not depend on f. What does C depend upon?

(d) Prove a similar result for |f(b)|2.

4. Consider the weak formulation of the steady state Dirichlet problem in the form: Find ) such that

,

where the bilinear form a(., .) and the linear functional f(.) are defined by

Z Z Z a(v, f) = k(x)v,if,i dV and f(f) = - gfdV - k(x)U˜,if,i dV.

V V V

Here k(x) is the non-constant conductivity, g(x) is a source term and U˜ is a H1(V ) extension of the boundary value U(x) to all of V .

(a) Suppose that there are constants m,M 0 such that for any x ? V , m = k(x) = M. Show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the norm. (Hint: Use the property of integrals that for any functions f(x) = h(x) then

Z Z

f dV = hdV.)

V V

(b) Assume that the source term g ? L2(V ). Show that the linear functional f(.) is bounded on .

(Hint: Use the result (1) from Question 1 above to bound the first integral in f(.).)

We have therefore shown that the assumptions of the Lax-Milgram Theorem are true for this problem.

5. Consider the weak formulation of the cooling problem described in Question 3(b) of Assignment 1: Findu ? H1(V ) such that a(u, f) = f(f) ?f ? H1(V ),

where the bilinear form a(., .) and linear functional f(.) are defined by

Z Z Z Z a(u, f) = k(x)u,if,i dV + hufdS and f(f) = - gfdV + hu0fdS.

V ?V V ?V

Here k(x) is the non-constant conductivity, g(x) is a source term, h(x) is the heat transfer coefficient on the boundary and u0(x) is the temperature of the external environment.

(a) With the same conditions on k as in Question 4(a) above, and suppose that there are constants hmin,hmax 0 such that hmax = h(x) = hmin, show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the H1 norm. (Hint: Use the general results (4) above (5) to handle the surface integral term in the bilinear form.)

(b) Assume that the source term g ? L2(V ). Show that the linear functional f(.) is bounded on H1(V ).

We have therefore shown that the assumptions of the Lax-Milgram Theorem are also true for this problem.

only 5 questions on file (Assignment 2.pdf )

Detailed explanation

Math 7731 – Mathematical Problems in Industry

Assignment 2 (2017)

1. In lectures we defined the Sobolev space ) with the norm

.

However, since ), it will also have the norm associated with the Sobolev space H1(V ),

.

We want to show that these are equivalent norms. That is, there are constants k,K 0 such that for any

),

. (1)

(a) Show that if (1) is true, then there also exists k0,K0 0 such that

. (2)

So equivalence is symmetric, as we would expect.

(b) We will show this equivalence for the one-dimensional case where V is the interval (a,b), though it is true for n = 2,3 dimensions also. The left hand inequality of (1) is obvious (with k = 1). Let f be any bounded and continuous function with bounded and continuous first derivatives, and which is zero on the boundary of V , that is at x = a and x = b. Then for any x ? (a,b),

where f0 = df/dx. Why?

(c) Apply the Cauchy-Schwarz inequality to the right hand side of this to obtain

.

(d) Show that

|f(x)|2 = ||f0||2L2|x - a| where ||.||L2 is the L2 norm defined in lectures.

(e) By integrating both sides with respect to x over the entire interval (a,b), conclude that

||f||2L2 = C||f0||2L2

for some C 0 which does not depend on f. What does C depend upon?

(f) Use the result (e) to prove the right hand inequality of (1).

(g) We have now proved (1) for the special case of f as described in (b). How do you think we would prove it for any )? (Describe in words.)

2. We want to prove the result that there exists M 0 such that for any f ? H1(V ),

. (3)

Intuitively, this result is saying that the size of f on V is “controlled” by the size of f on the boundary and the size of the first partial derivatives of f on V , where size is measured by the L2 norm. Again we will only consider the case where V is the interval (a,b), though the result is true for n = 2,3 dimensions also. In the one-dimensional case the boundary integral over ?V on the right hand side takes the simple form f(a)2 + f(b)2.

(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 1(b), show that or any x ? V ,

,

and so

(c) Integrate both sides with respect to x over the entire interval (a,b) to conclude that

where C does not depend on f. What does C depend upon?

(d) Deduce from (3) that there exists M0 0 such that for any f ? H1(V ),

. (4)

3. We next want to prove the result that there exists M 0 such that for any f ? H1(V ), the boundary integral

Z

f2 dS = M||f||2H1. (5)

?V

Intuitively, this is saying that the size of f on the boundary is controlled by the H1 norm of f on V . Again we will only consider the case where V is the interval (a,b), though the result is true for n = 2,3 dimensions also. In this one-dimensional case the boundary integral on the left hand side takes the simple form f(a)2 + f(b)2.

(a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 2(a), show that or any x ? V ,

,

and

(c) Integrate both sides with respect to x over the entire interval (a,b) to conclude that

|f(a)|2 = C||f||2H1

where C does not depend on f. What does C depend upon?

(d) Prove a similar result for |f(b)|2.

4. Consider the weak formulation of the steady state Dirichlet problem in the form: Find ) such that

,

where the bilinear form a(., .) and the linear functional f(.) are defined by

Z Z Z a(v, f) = k(x)v,if,i dV and f(f) = - gfdV - k(x)U˜,if,i dV.

V V V

Here k(x) is the non-constant conductivity, g(x) is a source term and U˜ is a H1(V ) extension of the boundary value U(x) to all of V .

(a) Suppose that there are constants m,M 0 such that for any x ? V , m = k(x) = M. Show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the norm. (Hint: Use the property of integrals that for any functions f(x) = h(x) then

Z Z

f dV = hdV.)

V V

(b) Assume that the source term g ? L2(V ). Show that the linear functional f(.) is bounded on .

(Hint: Use the result (1) from Question 1 above to bound the first integral in f(.).)

We have therefore shown that the assumptions of the Lax-Milgram Theorem are true for this problem.

5. Consider the weak formulation of the cooling problem described in Question 3(b) of Assignment 1: Findu ? H1(V ) such that a(u, f) = f(f) ?f ? H1(V ),

where the bilinear form a(., .) and linear functional f(.) are defined by

Z Z Z Z a(u, f) = k(x)u,if,i dV + hufdS and f(f) = - gfdV + hu0fdS.

V ?V V ?V

Here k(x) is the non-constant conductivity, g(x) is a source term, h(x) is the heat transfer coefficient on the boundary and u0(x) is the temperature of the external environment.

(a) With the same conditions on k as in Question 4(a) above, and suppose that there are constants hmin,hmax 0 such that hmax = h(x) = hmin, show that the energy norm (a(., .))1/2 derived from the bilinear form a(., .) is equivalent to the H1 norm. (Hint: Use the general results (4) above (5) to handle the surface integral term in the bilinear form.)

(b) Assume that the source term g ? L2(V ). Show that the linear functional f(.) is bounded on H1(V ).

We have therefore shown that the assumptions of the Lax-Milgram Theorem are also true for this problem.

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