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Unified and localized method and apparatus for solving linear and non-linear integral, integro-differential, and differential equations

  • US 20060111882A1
  • Filed: 10/03/2005
  • Published: 05/25/2006
  • Est. Priority Date: 11/23/2004
  • Status: Abandoned Application
First Claim
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1. A method of solving an Integro-Differential Equation (IDE) with an integral term having an integrand dependent on an integration variable α

  • , an independent variable x, a kernel function h′

    which depends on both x and α

    , and an unknown function f which is dependent on a single variable, said method comprising the steps of a. expressing said IDE in an equivalent Rao-X Integro-Differential Equation (ROXIDE) form wherein said integrand becomes dependent on f(x−

    α

    ) instead of f(α

    ), using, if necessary, the following two steps;

    i. finding a localized kernel function h of said kernel function h′

    in said equation using the General Rao Localization Transform; and

    ii. expressing said integral term in said IDE in a standard localized form of General Rao Transform using said localized kernel function h and said unknown function f, b. replacing f(x−

    α

    ) with a truncated Taylor-series expansion of f(x−

    α

    ) around x up to an integer order N, and setting all higher order terms to zero;

    c. replacing terms of said localized kernel function h dependent on x−

    α and

    α

    with its truncated Taylor series expansion around the point x and α

    ;

    d. simplifying the resulting expression by grouping terms based on the unknowns which are the derivatives of f with respect x at x denoted by f(n) for an n-th order derivative;

    moving the unknowns f(n) to be outside the definite integrals in integral terms that arise during simplification and grouping of terms;

    e. deriving a system of at least N equations by taking various derivatives with respect to x of the equation derived in Step (d), and setting to zero any derivatives of f of order greater than N to zero;

    computing symbolically or numerically, all definite integrals using the given value of x if needed, and obtaining a system of at least N equations; and

    f. Solving said system of at least N equations obtained in Step (e) to obtain the unknown f(0) and providing it as the desired solution f(x) of said IDE.

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