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The operational calculus for numerical solutions of differential equations and The Poly-processor.

[update Chap. 4 Sec. 1 (01/13/'15)]

. . This Home Page gives publicity to the author's work on the new method for the numerical solution of ordinary differential equations, the applications, the basic theories or the operational calculus of differential equations, and the hyper-parallel processor to which the author named Poly-processor.
. . The Poly-processor consists of a great many micro-processors which have the same micro-programs and are united to form in a simple pattern such as figure at the head of this page. It is analogous to the polymer which consists of a great many monomers and gets the property of fiber and plastic. The author draws the conclusion that the Poly-processor is analogous to the brain.
. . The operational calculus is the hyper-parallel calculus for numerical solutions of mathematical problems on the Poly-processor.
. . All these ideas originate with the author. The author submitted the first paper to IPSJ in 1977 and published "The operational calculus of numerical solutions and The Poly-processor" in 1994. This Home Page has some progress.

Copyright ©'77,'94,'06 Akio Wakabayashi.
All rights reserved.
Alias: PolyProcessor
E-mail: here


  1. Some troubles of numerical calculation.
  2. Preparing. Click Section 1, Section 2, Section 3 if it is possible to read japanese.

  3. Numerical solution of Differential equations and Applications.
    1. The new method for numerical solution of ordinary differential equations.
      1. Summary explanation of the operational calculus.
      2. The numerical solution of ordinary differential equations.
      3. Program and results.
      4. The Accuracy of solutions.
      5. The condition for existence of solutions.
      6. Stiff differential equations and usual troubles.
      7. Stability of solutions.
    2. Other methods introduced by the operational calculus.
      1. The methods using the polynomials of higher degree.
      2. Accuracy of the results by Program 2.1.
      3. The variable pitch method using 3 points.
      4. Results by the variable pitch method.
      5. The variable pitch method using 5 points.
      6. The results of the variable pitch method using 5 points.
      7. The definite integral by the numerical solution of differential equations.
    3. The Impulsive response of transfer function by differential equation.
      1. Introduction.
      2. Impulse function.
      3. Unit step function.
      4. Laplace transform of any function.
      5. Laplace transform and Fourier transform.
      6. Impulsive response and its differential equation.
      7. The numerical simulator of impulsive response.
      8. The problem having to be cautioned.
      9. The convolution integral.
      10. The differential equation and the transfer function with any excitation.
      11. Simplified method.
      12. The program of the numerical simulator.
      13. The most important idea.
    4. The systems of n-th order differential equations.
      1. The solution of the systems of n-th order differential equations.
      2. The numerical solution of closed-loop systems.
      3. The numerical simulator of closed-loop systems.

  4. The operational calculus.
    1. The vectors.
      1. The definitions of vectors.
      2. Some properties of the vector space.
      3. The finite differences of vectors.
      4. The product of vectors.
      5. The quotient of vectors.
      6. The expanded definition of the quotient of vectors.
      7. The vector-valued functions.
      8. The absolute value of the vector.
    2. The operators.
      1. The quotient without the vector form.
      2. The system of operators.
      3. The difference operator.
      4. The summation operator.
      5. The shifting operator.
      6. The operator-valued functions.
    3. Introducing the differential operator.
      1. Summary explanation of the differential operator.
      2. Newton's interpolation formula with remainder term.
      3. The difference of Newton's interpolation formula.
      4. Differentiation of Newton's interpolation formula.
      5. The differential operator.
      6. The second differential operator.
      7. The differential operator of higher order.
    4. Introducing the integral operator.
      1. Summary explanation of the integral operator.
      2. The integral of Newton's interpolation f rmula.
      3. The remainder of Simpson's 1/3 rule.
      4. The integral operator.
      5. The predictor and corrector.
      6. The double integral operator.
      7. The n multiple integral operator.
    5. The operator space.
      1. The vector space.
      2. The problems about the operator value and the norm.
      3. The definition of the norm of operators in usual differential and integral.
      4. The trouble of usual definition about continuous functions.
      5. The norm of operators of this operational calculus.

  5. Differential equations and applications.
    1. Differential equations of the first order.
      1. The method of solution by this operational calculus.
      2. The solution by symbolical formula manipulation.
      3. Lipschitz condition.
      4. The solution by interpolation and extrapolation.
      5. Taylor's expansion of vector valued function.
      6. Newton-Raphson's method.
      7. The formulas for differentiation and integration.
    2. Differential equations of the n-th order.
      1. Reducing the n-th order differential equation to the first order one.
      2. Some troubles of the n-th order differential equations.
      3. The solution by use of the operator valued function of the integral operator.
      4. The solution of the n-th order differential equation with constant coefficients.
      5. The n multiple integral by the solution of the first order differential equation.
      6. The condition which must be satisfied for the numerical solution to converge.
    3. Programming of the operational solutions of the n-th order differential equation.
      1. Denotation of the variables and constants.
      2. Some algorithms for the operational calculus.

    4. The differential analysis and the integral synthesis.
      1. Introducing an empirical formula by differentiation.
      2. Introducing the theoretical differential equation from the data by experiment.
      3. The frequency spectrum. by the differential analysis.
      4. Theoretical resonance.
    5. Expanding the operational calculus into that of two variables.
      1. The interpolation formula of two variables.
      2. Differentiation of the double vector.
      3. The integral of the double vector.
      4. The triple vector denoting the function of three variables.

  6. The poly-processor and poly-processing.
    1. The concept of the poly-processing.
      1. Similar thinking and point symmetric thinking.
      2. The point symmetric thinking in information processing.
      3. The point symmetric thinking in parallel processing and language.
      4. The parallel property of the operational calculus.
    2. The poly-processing system for the operational calculus.
      1. The poly-processing for addition and subtraction of vectors.
      2. Transform of function to vector and inverse transform.
      3. The poly-processing for multiplying two vectors.
      4. The poly-processing for calculating the quotient of vectors.
      5. The poly-processing for differentiating and integrating a vector.
      6. The poly-processing of multiplying the operators.
      7. The poly-processing of the quotient of matrices.
    3. Analyzing the construction of sentences and programming.
      1. Poly-processing of analyzing the construction of equations.

  7. The synthetic voice which can be easily synthesized.

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