To solve the heat equation with non-homogeneous boundary conditions, we transform the boundary conditions by subtracting the solution of the heat equation with homogeneous boundary conditions.
Added by Julian J.
Close
Step 1
We have the heat equation with non-homogeneous boundary conditions, which means the boundary conditions are not equal to zero. Show more…
Show all steps
Your feedback will help us improve your experience
Andrew Noble and 65 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Find the solution to the heat equation ut - c^2uxx = 0, c > 0, that satisfies the initial condition u(x,0) = f(x) and the boundary conditions ux(0,t) + k1u(0,t) = 0 = ux(L,t) + k2u(L,t), k1, k2 ∈ ℝ.
Adi S.
Sri K.
Prove the comparison principle for the heat equation: If u solves u_t = ku_{xx}, 0 < x < l, 0 < t < ∞ u(0,t) = g_1(t), u(l,t) = h_1(t) u(x,0) = ϕ_1(x) and v solves v_t = kv_{xx}, 0 < x < l, 0 < t < ∞ v(0,t) = g_2(t), v(l,t) = h_2(t) v(x,0) = ϕ_2(x) with g_1 ≤ g_2, h_1 ≤ h_2, and ϕ_1 ≤ ϕ_2, then u ≤ v for 0 ≤ t < ∞, 0 ≤ x ≤ l. (In other words, if u and v are two solutions of the heat equation and u ≤ v for the initial and boundary conditions, then u ≤ v everywhere).
Madhur L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD