ABSTRACT
E (x, y, t) = 1
4πat exp
( −x
4at
) .
◮ Formulas to construct particular solutions. Remarks on the Green’s functions.
1◦. Apart from usual separable solutions w(x, y, t) = f1(x)f2(y)f3(t), the equation in question has more sophisticated solutions in the product form
w(x, y, t) = u(x, t)v(y, t),
where u = u(x, t) and v = v(y, t) are solutions of the one-dimensional heat equations
∂u
∂t = a
∂2u
∂x2 ,
∂v
∂t = a
∂2v
∂y2 ,
considered in Section 3.1.1. 2◦. Suppose w = w(x, y, t) is a solution of the heat equation. Then the functions
w1 =Aw(±λx+C1, ±λy+C2, λ2t+C3), w2 =Aw(x cos β− y sin β+C1, x sin β+ y cos β+C2, t+C3), w3 =A exp
[ λ1x+λ2y+ a(λ
2 2)t ] w(x+2aλ1t+C1, y+2aλ2t+C2, t+C3),
w4 = A
δ+βt exp
[ − β(x
2+ y2)
4a(δ+βt)
] w
( x
δ+βt ,
y
δ+βt , γ+λt
δ+βt
) , λδ−βγ = 1,
where A, C1, C2, C3, β, δ, λ, λ1 and λ2 are arbitrary constants, are also solutions of this equation. The signs at λ’s in the formula for w1 are taken arbitrarily, independently of each other. ⊙ Literature: W. Miller, Jr. (1977). 3◦. In all two-dimensional boundary value problems discussed in Section 4.1.1, the Green’s function can be represented in the product form
G(x, y, ξ, η, t) = G1(x, ξ, t)G2(y, η, t),
where G1(x, ξ, t) and G2(y, η, t) are the Green’s functions of the corresponding one-dimensional boundary value problems (these functions are specified in Sections 3.1.1 and 3.1.2).
