# Numerical Solution of a Wide Class of Functional Problems

Original Publication Date: 1971-Feb-01

Included in the Prior Art Database: 2005-Feb-23

## Publishing Venue

IBM

## Related People

## Abstract

The numerical problem of determining the function y(p) which is solution of certain functional equations when the point p belongs to given domains of the space S(r) , is soved by a computational method based on substitution of the given domains by suitable finite and therefore discrete sets, which can be described point-by-point by means of a computer.

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__Page 1 of 5__**Numerical Solution of a Wide Class of Functional Problems **

The numerical problem of determining the function ϕ(ρ) which is solution of
certain functional equations when the point ρ belongs to given domains of the
space S_{(r)} , is soved by a computational method based on substitution of the
given domains by suitable finite and therefore discrete sets, which can be
described point-by-point by means of a computer.

The algorithm is particularly applicable to the solution of differential or integral problems whose boundary conditions are defined on sets of points which do not have any of the characteristics of regularity and symmetry needed by the ordinary methods. A typical example is mathematical models of water sheds, in which the differential equations have boundary conditions depending on the geographical and topographical arrangements of given areas.

Definition of the functional problem.

In a limited domain J_{0} of a Euclidean space S_{r} with r dimensions, let a
family J of functions f(p) of the point p of S_{r} be given. This family is assumed to
be a Banach space with norm

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Besides, let

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be µ given proper subset of J_{0} and

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be µ+i continuous functional operators, each of which gives a representation of J on itself; i.e. however we choose f(p)∈J, it is always

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Let us also suppose that the function ϕ(p)∈J, such that

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exists and is unique.

We say then that the search for function ϕ(p) which satisfies (5) is a functional problem:

The method of approximate numerical computation.

Let us consider a set V of a finite number N of points.

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1

__Page 2 of 5__belonging to S_{r} such that

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The (8) means that, however we choose a point P ∈ J_{0}, there will be at
least one point Pj ∈ V whose distance from P is not greater than .

Every function ψ(p) ∈ J and such that

(Image Omitted)

Functional problem (9) represents, with respect to problem(5), a form of
"discretization" that works on the domains J_{0},...,J_{µ} in which then vanishing of the
operators Ω_{i} is imposed.

This is convenient when we want to use a computer since the domains
J_{i}^{*}=( J_{i} ∩ V) may be entirely described point-to-point. Once we have defined the
domains J_{i}^{*}, there are infinite functions ψ(p) which are S-solutions of problem (5).

Determination of a S-solution of problem (5).

We indicate that { ψ_{m}(p) } a sequence of functions

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which have the following properties:

a) the functions ψ_{m}(p)= ψ_{m}(p; λ_{1},...,λ_{m}) depends on m parameters;

b) however we choose a function f(p)∈J, there exists a sequence λ_{1},λ_{2},...

such that

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Then, for a given value of m, let us introduce the functions

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and determine the value of parameters λ_{1},...,λ_{m} by imposing the conditions

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In this way (14) are a system on M finite equations with m unknowns λ_{1},...,λ_{m}

The minimum number M which makes the (14)'s solvable and which is certainly...