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• W2000680629 abstract "It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size <italic>h</italic> to: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon u Subscript x x Baseline plus b left-parenthesis x right-parenthesis u Subscript x Baseline equals f left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon {u_{xx}} + b(x){u_x} = f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than x greater-than 1 comma b greater-than 0> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0 > x > 1,b > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>b</italic> and <italic>f</italic> smooth, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=application/x-tex>varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in (0, 1], and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u left-parenthesis 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>u(0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u left-parenthesis 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>u(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C h squared> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>C{h^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the constant <italic>C</italic> independent of <italic>h</italic> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=application/x-tex>varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). This scheme was derived by El-Mistikawy and Werle by a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{C^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given." @default.
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• W2000680629 date "1981-01-01" @default.
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• W2000680629 title "An analysis of a uniformly accurate difference method for a singular perturbation problem" @default.
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• W2000680629 doi "https://doi.org/10.1090/s0025-5718-1981-0616361-0" @default.
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