Unified and localized method and apparatus for solving linear and non-linear integral, integro-differential, and differential equations
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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Abstract
This invention is based on a new class of mathematical transforms named Rao Transforms invented recently by the author of the present invention. Different types of Rao Transforms are used for solving different types of linear/non-linear, uni-variable/multi-variable integral/integro-differential equations/systems of equations. Methods and apparatus that are unified and computationally efficient are disclosed for solving such equations. These methods and apparatus are also useful in solving ordinary and partial differential equations as they can be converted to integral/integro-differential equations. The methods and apparatus of the present invention have applications in many fields including engineering, science, medicine, and economics.
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Citations
36 Claims
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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 ofa. 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. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34)
- , an independent variable x, a kernel function h′
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35. An apparatus for solving an integro-differential equation which includes:
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a. A means for reading as input an integro-differential equation with integral terms;
b. A means for applying General Rao Localization Transform to convert integral terms to General Rao Transform form and derive an integro-differential equation in ROXIDE form;
c. A means for truncated Taylor-series substitution and simplification of mathematical expressions derived from ROXIDEs;
d. A means for computing the derivatives of ROXIDEs and solving resulting algebraic equations to obtain a solution for said integro-differential equation; and
e. A means for providing the solution of said integro-differential equation as output. - View Dependent Claims (36)
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Current AssigneeMuralidhara Subbarao
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Original AssigneeMuralidhara Subbarao
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InventorsSubbarao, Muralidhara
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Application NumberUS11/242,191Publication NumberTime in Patent OfficeDaysField of SearchUS Class Current703/2CPC Class CodesG06F 17/13 Differential equations usin...
