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Integrodifferential equation which interpolates the heat equation and the wave equation. II. (English) Zbl 0796.45010

[For part I see ibid. 27, No. 2, 309-321 (1990; Zbl 0790.45009).]

We are concerned with the integrodifferential equation

(IDE) α u(t,x)=φ(x)+t α/2 Γ1 + α 2ψ(x)+1 Γ(α) 0 t (t-s) α-1 Δu(s,x)ds,t>0,x,

for 1α2. When ψ0, (IDE) 1 is reduced to the heat equation. For α=2, (IDE) 2 is just the wave equation and its solution u 2 (t,x) has the expression called d’Alembert’s formula:

u 2 (t,x)=1 2φ ( x + t ) + φ ( x - t )+1 2 x-t x+t ψ(y)dy·

The aim of the present paper is to investigate the structure of the solution of (IDE) α by its decomposition for every α, 1α2. We show that (IDE) α has the unique solution u α (t,x) (1α2) expressed as

u α (t,x)=1 2𝔼φ x + Y α (t) + φ x - Y α (t)+1 2𝔼 x-Y α (t) x+Y α (t) ψ(y)dy(1)

where Y α (t) is the continuous, nondecreasing and nonnegative stochastic process with Mittag-Leffler distribution of order α/2, and 𝔼 stands for the expectation. We remark that the expression (1) has the same form as that of d’Alembert’s formula.

MSC:
45K05Integro-partial differential equations
35K05Heat equation
35L05Wave equation (hyperbolic PDE)