diff --git a/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.pdf b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.pdf new file mode 100644 index 000000000..00e2b1e38 Binary files /dev/null and b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.pdf differ diff --git a/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.tm b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.tm new file mode 100644 index 000000000..bde6e564b --- /dev/null +++ b/docs/JacobiPolynomialSolutionForFractionalRiccatiDifferentialEquations.tm @@ -0,0 +1,175 @@ + + +> + +<\body> + \; + + + + Suppose =>, then for any + \\> (the set of all non-negative + integers), the weighted Sobolev space >|)>> + can be defined in the usual way, which indicates its inner product, + semi-norm and norm by + + <\equation> + >|)>> + + + <\equation> + >|)>> + + + and + + <\equation> + |v|\<\|\|\>>>|)>> + + + respectively. In particular, + + <\equation> + L|)>=H|)> + + + <\equation> + |v|\<\|\|\>>>=|v|\<\|\|\>>>> + + + and + + <\equation> + H>|)>=f,|f|\<\|\|\>>>\\|}> + + + \; + + <\equation> + |f|\<\|\|\>>>>=>>|\|>>|)>> + + + <\equation> + >>=|)>>|\|>> + + + Now we can suppose the function H>|)>> + in + + <\equation> + p,\>|)>=,\>,p,\>,\,p,\>|}> + + + as presented in the following formula: + + <\equation> + f=>k*p,\> + + + In which the coefficients > are generated by: + + <\equation> + k=>*p,\>*w,\>*d*x,i=0,\. + + + In practice, only the first -terms shifted Jacobi polynomials are + taken into account. Then we have: + + <\equation> + f=k*p,\>=K*P + + + with + + <\equation> + K=,k,\,k|]>,P=,\>,p,\>,\,p,\>|]> + + + In as much as ,\>> is a finite dimensional + vector space, has a unique best approximation from + ,\>>, say + \p,\>> that is: + + <\equation> + \y\p,\>,|f-f|\<\|\|\>>>\|f-y|\<\|\|\>>> + + + Guo and Wang (2004), came to the conclusion that for any + H>|)>>, + \\> and + \\\>, a generic positive constant + independent of any function, , > and > + exists so that: + + <\equation> + |f-f|\<\|\|\>>>,\>\C**\*+\+1|)>|)>>>> + + + + + We can express Riemann-Liouville fractional integral operator of order + > of the vector by: + + <\equation> + I>*P\Q|)>>*P + + + where |)>>> is the n> + operational matrix of Riemann-Liouville fractional integral of order + >. + + + + If |)>>> is the n> operational + matrix of Riemann-Liouville fractional integral of order >, + then the elements of this matrix are taken as: + + <\equation> + Q|)>>=|)>>|}>=P|)>>***B+\+1,\+1|)>|]> + + + Now, we define the error vector , as + + <\equation> + E=I>*P-Q|)>>*P + + + +<\initial> + <\collection> + + + + + + + + +<\references> + <\collection> + > + > + > + + + +<\auxiliary> + <\collection> + <\associate|toc> + |math-font-series||The + approximation of functions in the Sobolev space> + |.>>>>|> + + + |math-font-series||The + operational matrix of fractional integral> + |.>>>>|> + + + |Theorem 3.1. + |.>>>>|> + > + + + \ No newline at end of file