(k) The solution of the following problem for the heat equation with nonhomge- neous boundary conditions: U4₂=Uzz 00, u(0, t) = 0, u(1, t) = 1, t≥0, u(1,0) = (x), OSI≤1. can be obtained by letting u(x, t) = v(r, t) + g(x), where g(z) satisfies: (A) g"(x) = -1, g(0) = g(1) = 0 (B) g'(x) = -1, g(0) = 0, g(1) = 1 (C) g'(x) = 0, g(0) = 0, g(1) = 1 (D) g'(x) = 0, 9(0) = 0, 9(1) = -1
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- 5.(4) Suppose z = f(x, y) where f is differentiable (both f, and fy exist). In addition, suppose = g(t) and y=h(t), and I= dz Find when t-3. dt g(3) = 2, g'(3) = 5, fr(2,7)=6, h(3) = 7, h'(3) = -4, fy(2, 7) = −8.2. Consider the heat equation ди k- for 0 0 with the boundary conditions u(0, t) = 0 and u(L, t) = 0, for t > 0. Solve for the following initial value conditions: a) f(x) = 4 sin ( 1 b) f(x) = x € (0, L/2] x € (L/2, L)
- 1. Consider the heat equation ди = k for - T 0 | with the periodic boundary conditions u(-7, t) = u(1, t) and du (-7,t) = du (7T, t), for t > 0, and the initial value condition u(x, 0) = f(x) for x E [-7, 7]. Solve for the following initial value conditions: a) f(x) = x So re (-7,0]| x € а€ (0, т) b) f(x) =do Q1 2 ONLYM3
- 4.2 Find the solution subject to the following boundary and initial conditions of the heat equation, ut = a²uxx- (b) u(0, t) = u(2,t) = 0, u(r,0) = 1 if 0Help me understand bettera?u subject to boundary conditions u(0,t)= u(2,t) = 0 du Q3/ Determine the solution of %3D at ax2 and the initial condition u(x,0) = x, where 2 is the length of the bar.1 ' The solution of the heat equation wzz =wt, 0Q4 (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 02. Find the solution to the following non-homogeneous heat equation with given initial and boundary conditions on [0, π]. Ut Uxx = e -2t sin (3x), u(0, t) = u(π, t) = 0, u(x,0) = sinx.SEE MORE QUESTIONSRecommended 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,