Applied Mathematical Sciences

I. Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH).- I. Introduction of the Operator h Through the Convolution Ring C.- 1. Convolution Ring.- 2. Operator of Integration h.- II. Introduction of the Operator s Through the Ring CH.- 3. The Ring CH and the Identity Operator I = h/h.- 4. CH as a Class of Generalized Functions of Hyperfunctions.- 5. Operator of Differentiation s and Operator of Scalar Multiplication [?].- 6. The Theorem $$frac{I}{{s - [alpha ]}} = {e^{alpha t}}$$.- III. Linear Ordinary Differential Equations with Constant Coefficients.- 7. The Conversion of the Initial Value Problem for the Differential Equation into a Hyperfunction Equation.- 8. The Polynomial Ring of Polynomials in s has no Zero Factors.- 9. The Partial Fraction Decomposition of a Rational Function of s.- 10. Hyperfunction Solution of the Ordinary Differential Equation (The Operational Calculus).- 11. Boundary Value Problems for Ordinary Differential Equations.- IV. Fractional Powers of Hyperfunctions h, s and $$frac{I}{{S - alpha }}$$.- 12. Euler's Integrals - The Gamma Function and Beta Function.- 13. Fractional Powers of h, of (s-?)?1, and of (s-?).- V. Hyperfunctions Represented by Infinite Power Series in h.- 14. The Binomial Theorem.- 15. Bessel's Function Jn(t).- 16. Hyperfunctions Represented by Power Series in h.- II. Linear Ordinary Differential Equations with Linear Coefficients (The Class C/C of Hyperfunctions).- VI. The Titchmarsh Convolution Theorem and the Class C/C.- 17. Proof of the Titchmarsh Convolution Theorem.- 18. The Class C/C of Hyperfunctions.- VII. The Algebraic Derivative Applied to Laplace's Differential Equation.- 19. The Algebraic Derivative.- 20. Laplace's Differential Equation.- 21. Supplements. I: Weierstrass' Polynomial Approximation Theorem. II: Mikusi?ski's Theorem of Moments.- III. Shift Operator exp(??s) and Diffusion Operator exp(??s1/2).- VIII. Exponential Hyperfunctions exp(??s) and exp(??s1/2).- 22. Shift Operator exp(??s) = e??sFunction Space K = K[0,?).- 23 Hyperfunction-Valued Function f(?) and Generalized Derivative $$frac{d}{{dlambda }}fleft( lambda right) = f'left( lambda right)$$.- 24. Exponential Hyperfunction exp(?s)=e?s.- 25. Examples of Generalized Limit. Power Series in e?s.- $$int_{0}^{infty } {{{e}^{{ - lambda s}}}} fleft( lambda right)dlambda = left{ {fleft( t right)} right}Forleft{ {fleft( t right)} right} in C$$.- 27. Logarithmic Hyperfunction w and Exponential Hyperfunction exp $$ left( { - lambda {{s}^{{1/2}}}} right) = {{e}^{{ - lambda {{s}^{{{{1} left/ {2} right.}}}}}}} $$.- 28. Logarithmic Hyperfunction w and Exponential Hyperfunction exp(?w).- IV. Applications to Partial Differential Equations.- IX. One DimensionaL Wave Equation.- 29. Hyperfunction Equation of the form f?(?) - w2f(?) = g(?), w ? C/C.- 30. The Vibration of a String.- 31. D'Alembert's Method.- 32. The Vibration of an Infinitely Long String.- X. Telegraph Equation.- 33. The Hyperfunction Equation of the Telegraph Equation.- 34. A Cable With Infinitely Small Loss.- X. (cont.).- 35. Conductance Without Deformation.- 36. The Thomson Cable.- 37. Concrete Representations of exp $$left( { - lambda sqrt {alpha s + beta } } right) $$.- 38. A Cable without Self-Induction.- 39. A Cable without Leak-Conductance.- 40. The Case Where All the Four Parameters Are Positive.- Positive.- XI. Heat Equation.- 41. The Temperature of a Heat-Conducting Bar.- 42. An Infinitely Long Bar.- 43. A Bar Without an Outgoing Flow of Heat.- 44. The Temperature in a Bar with a Given Initial Temperature.- 45. A Heat-Conducting Ring.- 46. Non-Insulated Heat Conduction.- Answers to Exercises.- Formulas and Tables.- References.- Propositions and Theorems in Sections.