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Zbl 0796.45010
Fujita, Yasuhiro
Integrodifferential equation which interpolates the heat equation and the wave equation. II. (English)
[J] Osaka J. Math. 27, No.4, 797-804 (1990). ISSN 0030-6126

[For part I see ibid. 27, No. 2, 309-321 (1990; Zbl 0790.45009).]\par We are concerned with the integrodifferential equation $$\text {(IDE)}\sb \alpha \quad u(t,x) = \varphi (x) + {t\sp{\alpha/2} \over \Gamma \Bigl( 1+ {\alpha \over 2} \Bigr)} \psi (x) + {1 \over \Gamma (\alpha)} \int\sp t\sb 0 (t-s)\sp{\alpha-1} \Delta u (s,x)ds, \quad t>0,\ x \in \bbfR,$$ for $1 \le \alpha \le 2$. When $\psi \equiv 0$, $\text {(IDE)}\sb 1$ is reduced to the heat equation. For $\alpha=2$, $\text {(IDE)}\sb 2$ is just the wave equation and its solution $u\sb 2(t,x)$ has the expression called d'Alembert's formula: $$u\sb 2(t,x) = {1 \over 2} \bigl[ \varphi (x+t)+ \varphi (x-t) \bigr] + {1 \over 2} \int\sp{x+t}\sb{x-t} \psi (y)dy.$$ The aim of the present paper is to investigate the structure of the solution of $\text {(IDE)}\sb \alpha$ by its decomposition for every $\alpha$, $1 \le \alpha \le 2$. We show that $(\text {IDE})\sb \alpha$ has the unique solution $u\sb \alpha (t,x)$ $(1 \le \alpha \le 2)$ expressed as $$u\sb \alpha (t,x) = {1 \over 2} \bbfE \biggl[ \varphi \bigl( x+Y\sb \alpha (t) \bigr) + \varphi\bigl(x - Y\sb \alpha(t)\bigr)\biggr] + {1 \over 2} \bbfE \int\sp{x+Y\sb \alpha (t)}\sb{x-Y\sb \alpha (t)} \psi (y) dy \tag 1 $$ where $Y\sb \alpha(t)$ is the continuous, nondecreasing and nonnegative stochastic process with Mittag-Leffler distribution of order $\alpha/2$, and $\bbfE$ stands for the expectation. We remark that the expression (1) has the same form as that of d'Alembert's formula.

MSC 2000:: *45K05 Integro-partial differential equations
35K05 Heat equation
35L05 Wave equation

Keywords: solution in explicit form; heat equation; wave equation; d'Alembert's formula

Citations: Zbl 0790.45009

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