The approximation of functions in the Sobolev space

crowlogic · Jan 20, 2024 · f92ada2 · f92ada2
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diff --git a/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.pdf b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.pdf
diff --git a/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.tm b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.tm
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+<TeXmacs|2.1.1>
+
+<style|<tuple|generic|alt-colors|boring-white|framed-theorems>>
+
+<\body>
+  \;
+
+  <section*|The approximation of functions in the Sobolev space>
+
+  Suppose <math|\<Omega\>=<around|[|0,1|]>>, then for any
+  <math|\<gamma\>\<in\>\<bbb-N\><rsub|0>> (the set of all non-negative
+  integers), the weighted Sobolev space <math|H<rsub|w><rsup|\<gamma\>><around|(|\<Omega\>|)>>
+  can be defined in the usual way, which indicates its inner product,
+  semi-norm and norm by
+
+  <\equation>
+    <around|(|u,v|)><rsub|w,\<Omega\>><rsup|<around|(|\<gamma\>|)>>
+  </equation>
+
+  <\equation>
+    <around|\||v|\|><rsub|w,\<Omega\>><rsup|<around|(|\<gamma\>|)>>
+  </equation>
+
+  and
+
+  <\equation>
+    <around|\<\|\|\>|v|\<\|\|\>><rsub|w,\<Omega\>><rsup|<around|(|\<gamma\>|)>>
+  </equation>
+
+  respectively. In particular,
+
+  <\equation>
+    L<rsub|2><around|(|\<Omega\>|)>=H<rsub|w><rsup|0><around|(|\<Omega\>|)>
+  </equation>
+
+  <\equation>
+    <around|\<\|\|\>|v|\<\|\|\>><rsub|w,\<Omega\>>=<around|\<\|\|\>|v|\<\|\|\>><rsub|w,\<Omega\>><rsup|<around|(|0|)>>
+  </equation>
+
+  and
+
+  <\equation>
+    H<rsub|w><rsup|\<gamma\>><around|(|\<Omega\>|)>=<around|{|f\<mid\>f<text|can
+    be measured>,<around|\<\|\|\>|f|\<\|\|\>><rsub|w,\<Omega\>>\<less\>\<infty\>|}>
+  </equation>
+
+  \;
+
+  <\equation>
+    <around|\<\|\|\>|f|\<\|\|\>><rsub|w,\<Omega\>><rsup|\<gamma\>>=<sqrt|<around*|(|<big|sum><rsub|i=0><rsup|\<infty\>><around|\||f<rsup|<around|(|i|)>>|\|><rsup|2><rsub|w,\<Omega\>>|)>>
+  </equation>
+
+  <\equation>
+    <around|\||f|\|><rsub|w,\<Omega\>><rsup|\<gamma\>>=<around|\||f<rsup|<around|(|\<gamma\>|)>>|\|><rsub|w,\<Omega\>>
+  </equation>
+
+  Now we can suppose the function <math|f\<in\>H<rsub|w><rsup|\<gamma\>><around|(|\<Omega\>|)>>
+  in
+
+  <\equation>
+    p<rsub|m,\<alpha\>,\<beta\>><around|(|\<Omega\>|)>=<text|span><around|{|p<rsub|0><rsup|\<alpha\>,\<beta\>><around|(|x|)>,p<rsub|1><rsup|\<alpha\>,\<beta\>><around|(|x|)>,\<ldots\>,p<rsub|m><rsup|\<alpha\>,\<beta\>><around|(|x|)>|}><text|>
+  </equation>
+
+  as presented in the following formula:
+
+  <\equation>
+    f<around|(|x|)>=<big|sum><rsub|i=0><rsup|\<infty\>>k<rsub|i>*p<rsub|i><rsup|\<alpha\>,\<beta\>><around|(|x|)><text|>
+  </equation>
+
+  In which the coefficients <math|k<rsub|i>> are generated by:
+
+  <\equation>
+    k<rsub|i>=<frac|1|g<rsub|i><rsup|2>>*<big|int><rsub|0><rsup|1>p<rsub|i><rsup|\<alpha\>,\<beta\>><around|(|x|)>*w<rsub|\<alpha\>,\<beta\>><around|(|x|)>*d*x,<space|1em>i=0,\<ldots\>.<text|>
+  </equation>
+
+  In practice, only the first <math|m>-terms shifted Jacobi polynomials are
+  taken into account. Then we have:
+
+  <\equation>
+    f<around|(|x|)>=<big|sum><rsub|i=0><rsup|m-1>k<rsub|i>*p<rsub|i><rsup|\<alpha\>,\<beta\>><around|(|x|)>=K<rsup|T>*P<text|>
+  </equation>
+
+  with
+
+  <\equation>
+    K=<around|[|k<rsub|0>,k<rsub|1>,\<ldots\>,k<rsub|m-1>|]><rsup|T>,<space|1em>P=<around|[|p<rsub|0><rsup|\<alpha\>,\<beta\>><around|(|x|)>,p<rsub|1><rsup|\<alpha\>,\<beta\>><around|(|x|)>,\<ldots\>,p<rsub|m-1><rsup|\<alpha\>,\<beta\>><around|(|x|)>|]><text|>
+  </equation>
+
+  In as much as <math|p<rsub|m,\<alpha\>,\<beta\>>> is a finite dimensional
+  vector space, <math|f> has a unique best approximation from
+  <math|p<rsub|m,\<alpha\>,\<beta\>>>, say
+  <math|f<rsub|m><around|(|x|)>\<in\>p<rsub|m,\<alpha\>,\<beta\>>> that is:
+
+  <\equation>
+    \<forall\>y\<in\>p<rsub|m,\<alpha\>,\<beta\>>,<around|\<\|\|\>|f<around|(|x|)>-f<rsub|m><around|(|x|)>|\<\|\|\>><rsub|w,\<Omega\>>\<leq\><around|\<\|\|\>|f<around|(|x|)>-y|\<\|\|\>><rsub|w,\<Omega\>><text|>
+  </equation>
+
+  Guo and Wang (2004), came to the conclusion that for any
+  <math|f\<in\>H<rsub|w><rsup|\<gamma\>><around|(|\<Omega\>|)>>,
+  <math|\<gamma\>\<in\>\<bbb-N\><rsub|0>> and
+  <math|0\<leq\>\<mu\>\<leq\>\<gamma\>>, a generic positive constant <math|C>
+  independent of any function, <math|m>, <math|\<alpha\>> and <math|\<beta\>>
+  exists so that:
+
+  <\equation>
+    <around|\<\|\|\>|f<around|(|x|)>-f<rsub|m><around|(|x|)>|\<\|\|\>><rsub|w,\<Omega\>><rsup|\<gamma\>,\<mu\>>\<leq\>C*<around|(|<around|(|m-1|)>*\<Gamma\>*<around|(|\<alpha\>+\<beta\>+1|)>|)><rsup|-\<mu\>><frac|1|2><around|\||f|\|><rsub|w,\<Omega\>><rsup|\<gamma\>><text|>
+  </equation>
+
+  <section*|The operational matrix of fractional integral>
+
+  We can express Riemann-Liouville fractional integral operator of order
+  <math|\<mu\>> of the vector by:
+
+  <\equation>
+    I<rsup|\<mu\>>*P\<approx\>Q<rsup|<around|(|\<mu\>|)>>*P<text|>
+  </equation>
+
+  where <math|Q<rsup|<around|(|\<mu\>|)>>> is the <math|m\<times\>n>
+  operational matrix of Riemann-Liouville fractional integral of order
+  <math|\<mu\>>.
+
+  <subsection*|Theorem 3.1.>
+
+  If <math|Q<rsup|<around|(|\<mu\>|)>>> is the <math|m\<times\>n> operational
+  matrix of Riemann-Liouville fractional integral of order <math|\<mu\>>,
+  then the elements of this matrix are taken as:
+
+  <\equation>
+    Q<rsup|<around|(|\<mu\>|)>><rsub|i,j>=<around*|{|q<rsub|i,j><rsup|<around|(|\<mu\>|)>>|}><rsub|i,j=0><rsup|n-1>=<big|sum><rsub|k=0><rsup|i><binom|i|k>P<rsub|i-k><rsup|<around|(|\<mu\>|)>><around|(|j|)>*<around*|[|\<Gamma\>*<around|(|k+1|)>*B<around|(|k+l+\<mu\>+\<beta\>+1,\<alpha\>+1|)>|]><text|>
+  </equation>
+
+  Now, we define the error vector <math|E>, as
+
+  <\equation>
+    E=I<rsup|\<mu\>>*P-Q<rsup|<around|(|\<mu\>|)>>*P<text|>
+  </equation>
+</body>
+
+<\initial>
+  <\collection>
+    <associate|magnification|1.2>
+    <associate|page-height|auto>
+    <associate|page-medium|paper>
+    <associate|page-type|letter>
+    <associate|page-width|auto>
+  </collection>
+</initial>
+
+<\references>
+  <\collection>
+    <associate|auto-1|<tuple|?|1|../../.TeXmacs/texts/scratch/no_name_18.tm>>
+    <associate|auto-2|<tuple|15|2|../../.TeXmacs/texts/scratch/no_name_18.tm>>
+    <associate|auto-3|<tuple|16|2|../../.TeXmacs/texts/scratch/no_name_18.tm>>
+  </collection>
+</references>
+
+<\auxiliary>
+  <\collection>
+    <\associate|toc>
+      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|The
+      approximation of functions in the Sobolev space>
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-1><vspace|0.5fn>
+
+      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|The
+      operational matrix of fractional integral>
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-2><vspace|0.5fn>
+
+      <with|par-left|<quote|1tab>|Theorem 3.1.
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-3>>
+    </associate>
+  </collection>
+</auxiliary>