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Recurrence formula for some higher order evolution equations

Yoritaka Iwata

Abstract

Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. %%% In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.

Recurrence formula for some higher order evolution equations

Abstract

Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. %%% In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.
Paper Structure (10 sections, 3 theorems, 70 equations)

This paper contains 10 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Mathematical settings
  3. Back ground and basic concepts
  4. Logarithmic representation of operators
  5. Miura transform
  6. Main result
  7. Recurrence formula generalizing the combined Miura transform
  8. Logarithmic representation of $n$-th order infinitesimal generator
  9. $n$-th order generalization of Hille-Yosida type exponential function of operator
  10. summary

Key Result

Theorem 4.1

For $t \in [0,T]$, let generally-unbounded closed operators $A_1(t), A_2(t), \cdots A_n(t)$ be continuous with respect to $t$ defined in a Banach space $X$. Here $A_1(t)$ is further assumed to be an infinitesimal generator of the first-order evolution equation in $X$, where $\psi(t)$ satisfying the initial condition $\psi(0) = \psi_0 \in X$ is the solution of the Cauchy problem of cp1. The commut

Theorems & Definitions (6)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof