Q4 (a) The temperature distribution u(x, t) of the one-dimensional gold rod is governed by the heat equation as follows. a²u 0.25 əx² ди at Given the boundary conditions u(0, t) = 2t?, u(1, t) = 5t, for 0
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- Laplace theorem of discontinious equation.20. Let g(t) be a differentiable function such that g(2) = 3, g(4) = −2, g'(2) = −1, and g'(4) = 5. Let F+ dF represent the differential (linear approximation) of F(2.05, 2.10, 4.10), where F(x, y, z) = ex². · g(x² − ye²−xy). Let II be the plane tangent to the implicit surface F(x, y, z) = 3 at the point (2,2,4). Let z-int represent the z-intercept of II. Find F + dF and z-int.Determine p" (xo), p" (xo) and p" (xo) for the given point xo if y = p(x) is a solution of the given initial value problem. y" + xy' +y = 0, y(0) = 2, y' (0) = 1 f" (0) : i ф" (0) - i pi" (0) = '(0): ||1 ' The solution of the heat equation wzz =wt, 0(b) Construct the Green's function of the problem: y" + y' - 2y = r(x), 0Sx<1 y(0) = y'(1) = 0 and hence solve the boundary value problem for r(x) = 1.Using the initial and boundary conditions solve Uxx- Uu = 0, c2 where initial conditions are and boundary conditions are U(x,0) = 0, U (x,0) = f(x) . U (0, t) = 0, U (X, t) = 0 where 0Let u be a solution of the heat equation u, -Uxx ; 00 u(0,t)=u(r,t)=0 u(x,0)= sin x+ sin 2x; 00 Then. 1: (a) u(x,t)→0 as t→o for all x e (0,7) (b) t'u(x,t)→0 as t→o for all xe (0, (c) - e'u(x,t) is a bounded function for xe (0,7),t >0 (d) e"u(x,t)→0 as t→o for all xe (0,7)Find the steady-state temperature u(r,θ) in a circular hole in an infinite plate of radius r=1 subject to the condition u(1,θ) = {π-θ, 0<θ<π and θ-π, π<θ<2π}y(t) Find a solution to the initial value problem that is continuous on the interval [0, 2π] where sin(t) g(t) = { - sin(t) = y' + sin(t)y = g(t), y(0) = 7, 1 if 0 ≤ t ≤ π, if π < t < 2π. if 0 ≤ t ≤ π, if π < t < 2π.(а) The temperature distribution u(x, t) of the one-dimensional gold rod is governed by the heat equation as follows. Q4 a²u ди -= 0.25 at Given the boundary conditions u(0, t) = 2t², u(1, t) = 5t, for 0Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,