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Volume 1, Issue 2
Estimation for Solutions of Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators

Guan-Quan Zhang

J. Comp. Math., 1 (1983), pp. 148-160.

Published online: 1983-01

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  • Abstract

In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation$\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$, where A(t) is a p. d. o. (pseudo-differential operator(s)) of order 1 or 2, N(t) is a uniformly bounded $H-›H$ linear operator. It is proved that if the symbol of the principal part of A(t) satisfies certain algebraic conditions, two estimates for the solution u(t) hold. One is similar to the estimate for analytic functions in the Three-Circle Theorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$.

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@Article{JCM-1-148, author = {Guan-Quan Zhang}, title = {Estimation for Solutions of Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators}, journal = {Journal of Computational Mathematics}, year = {1983}, volume = {1}, number = {2}, pages = {148--160}, abstract = {

In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation$\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$, where A(t) is a p. d. o. (pseudo-differential operator(s)) of order 1 or 2, N(t) is a uniformly bounded $H-›H$ linear operator. It is proved that if the symbol of the principal part of A(t) satisfies certain algebraic conditions, two estimates for the solution u(t) hold. One is similar to the estimate for analytic functions in the Three-Circle Theorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9691.html} }
TY - JOUR T1 - Estimation for Solutions of Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators AU - Guan-Quan Zhang JO - Journal of Computational Mathematics VL - 2 SP - 148 EP - 160 PY - 1983 DA - 1983/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9691.html KW - AB -

In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation$\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$, where A(t) is a p. d. o. (pseudo-differential operator(s)) of order 1 or 2, N(t) is a uniformly bounded $H-›H$ linear operator. It is proved that if the symbol of the principal part of A(t) satisfies certain algebraic conditions, two estimates for the solution u(t) hold. One is similar to the estimate for analytic functions in the Three-Circle Theorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$.

Guan-Quan Zhang. (1983). Estimation for Solutions of Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators. Journal of Computational Mathematics. 1 (2). 148-160. doi:
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