## 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 $\text{(IDE)}_ \alpha \quad u(t,x) = \varphi (x) + {t^{\alpha/2} \over \Gamma \Bigl( 1+ {\alpha \over 2} \Bigr)} \psi (x) + {1 \over \Gamma (\alpha)} \int^ t_ 0 (t-s)^{\alpha-1} \Delta u (s,x)ds, \quad t>0,\;x \in \mathbb{R},$ for $$1 \leq \alpha \leq 2$$. When $$\psi \equiv 0$$, $$\text{(IDE)}_ 1$$ is reduced to the heat equation. For $$\alpha=2$$, $$\text{(IDE)}_ 2$$ is just the wave equation and its solution $$u_ 2(t,x)$$ has the expression called d’Alembert’s formula: $u_ 2(t,x) = {1 \over 2} \bigl[ \varphi (x+t)+ \varphi (x-t) \bigr] + {1 \over 2} \int^{x+t}_{x-t} \psi (y)dy.$ The aim of the present paper is to investigate the structure of the solution of $$\text{(IDE)}_ \alpha$$ by its decomposition for every $$\alpha$$, $$1 \leq \alpha \leq 2$$. We show that $$(\text{IDE})_ \alpha$$ has the unique solution $$u_ \alpha (t,x)$$ $$(1 \leq \alpha \leq 2)$$ expressed as $u_ \alpha (t,x) = {1 \over 2} \mathbb{E} \biggl[ \varphi \bigl( x+Y_ \alpha (t) \bigr) + \varphi\bigl(x - Y_ \alpha(t)\bigr)\biggr] + {1 \over 2} \mathbb{E} \int^{x+Y_ \alpha (t)}_{x-Y_ \alpha (t)} \psi (y) dy \tag{1}$ where $$Y_ \alpha(t)$$ is the continuous, nondecreasing and nonnegative stochastic process with Mittag-Leffler distribution of order $$\alpha/2$$, and $$\mathbb{E}$$ stands for the expectation. We remark that the expression (1) has the same form as that of d’Alembert’s formula.

### MSC:

 45K05 Integro-partial differential equations 35K05 Heat equation 35L05 Wave equation

Zbl 0790.45009