Operational Calculus

1. Front Matter

   Pages i-x

   PDF

2. Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH)

   1. Introduction of the Operator h Through the Convolution Ring C

      • K. Yosida

      Pages 1-4

   2. Introduction of the Operator s Through the Ring CH

      • K. Yosida

      Pages 5-13

   3. Linear Ordinary Differential Equations with Constant Coefficients

      • K. Yosida

      Pages 14-31

   4. Fractional Powers of Hyperfunctions h, s and \( \frac{I}{{S - \alpha }} \)

      • K. Yosida

      Pages 32-38

   5. Hyperfunctions Represented by Infinite Power Series in h

      • K. Yosida

      Pages 39-46

3. Linear Ordinary Differential Equations with Linear Coefficients (The Class C/C of Hyperfunctions)

   1. The Titchmarsh Convolution Theorem and the Class C/C

      • K. Yosida

      Pages 47-52

   2. The Algebraic Derivative Applied to Laplace’s Differential Equation

      • K. Yosida

      Pages 53-73

4. Shift Operator exp(−λs) and Diffusion Operator exp(−λs1/2)

   1. Exponential Hyperfunctions exp(−λs) and exp(−λs1/2)

      • K. Yosida

      Pages 74-105

5. Applications to Partial Differential Equations

   1. Front Matter

      Pages 106-107

      PDF

   2. One Dimensional Wave Equation

      • K. Yosida

      Pages 108-123

   3. Telegraph Equation

      • K. Yosida

      Pages 124-144

   4. Heat Equation

      • K. Yosida

      Pages 145-156

6. Back Matter

   Pages 157-171

   PDF

In the end of the last century, Oliver Heaviside inaugurated an operational calculus in connection with his researches in electromagnetic theory. In his operational calculus, the operator of differentiation was denoted by the symbol "p". The explanation of this operator p as given by him was difficult to understand and to use, and the range of the valid­ ity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general. In the 1930s, Gustav Doetsch and many other mathematicians began to strive for the mathematical foundation of Heaviside's operational calculus by virtue of the Laplace transform -pt e f(t)dt. ( However, the use of such integrals naturally confronts restrictions con­ cerning the growth behavior of the numerical function f(t) as t ~ ~. At about the midcentury, Jan Mikusinski invented the theory of con­ volution quotients, based upon the Titchmarsh convolution theorem: If f(t) and get) are continuous functions defined on [O,~) such that the convolution f~ f(t-u)g(u)du =0, then either f(t) =0 or get) =0 must hold. The convolution quotients include the operator of differentiation "s" and related operators. Mikusinski's operational calculus gives a satisfactory basis of Heaviside's operational calculus; it can be applied successfully to linear ordinary differential equations with constant coefficients as well as to the telegraph equation which includes both the wave and heat equa­ tions with constant coefficients.

• Derivative
• Finite
• Hyperfunktion
• Identity
• Operatorenrechnung
• algebra
• calculus
• differential equation
• equation
• function
• logarithm
• proof
• theorem

• Kamakura 247, Japan

  K. Yosida

Policies and ethics