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.


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


Zbl 0790.45009