# Numerical solution of two point boundary value problems by Herbert Bishop Keller

By Herbert Bishop Keller

Lectures on a unified conception of and functional strategies for the numerical answer of very common periods of linear and nonlinear aspect boundary-value difficulties.

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0 else ⎩ Besides their use in what is to come, fuzzy operators have been used extensively in multicriteria decision making [Bellman and Zadeh, 1970; Yager, 1988; Yager, 2004]. We end this section with an example of the use of fuzzy set operators in multicriteria decision making. 1. 4, to assess cancer risk based on the following observations. Suppose we decide that cancer risk should be high if either internal factors or environmental factors are high. This is modeled by a union operator (OR).

The limits as w → ∞ , generate the drastic union and intersection, defined by ⎧ A( x) if B( x) = 0 ⎪ ( A ∪d B)( x) = ⎨ B( x) if A( x) = 0 and ⎪ 1 else ⎩ ⎧ A( x) if B( x) = 1 ⎪ ( A ∩d B)( x) = ⎨ B( x) if A( x) = 1 . ⎪ 0 else ⎩ Besides their use in what is to come, fuzzy operators have been used extensively in multicriteria decision making [Bellman and Zadeh, 1970; Yager, 1988; Yager, 2004]. We end this section with an example of the use of fuzzy set operators in multicriteria decision making. 1.

An alternate formulation, denoted as a Takagi-Sugeno-Kang (TKS) system [Takagi and Sugeno, 1985; Sugeno and Kang, 1988] only modifies the membership functions in the consequent clause. It was developed for control applications where the output of the rule firing should be a function of the set of crisp input values. Instead of a general fuzzy set B of Y, the output of each rule is a specific function of the real inputs. The antecedent part of each rule, Ri, is matched as in the MA approach, but the output then becomes yi = Ai′1 × × Ain′ ( x1 ,…, xn ) ⋅ f i ( x1 ,…, xn ) .