Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 1 1 2014 08 01 Application of Shannon wavelet for solving boundary value problems of fractional differential equations I 33 42 6349 EN K. Nouri School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: 19395-5746, Tehran, Islamic Republic of Iran N. Bahrami Siavashani Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box: 35195-363, Semnan, Islamic Republic of Iran. Journal Article 2014 07 01 <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erential equations. Therefore, a reliable and e<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ffi<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">cient technique as a solution is regarded. <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">This paper develops approximate solutions for boundary value problems of <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erential equations with non-integer order by using the Shannon wavelet <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">bases. Wavelet bases have di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erent resolution capability for approximating <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">of di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erent functions. Since for Shannon-type wavelets, the scaling function <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">and the mother wavelet are not necessarily absolutely integrable, the partial <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">sums of the wavelet series behave di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erently and a more stringent condition, <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">such as bounded variation, is needed for convergence of Shannon wavelet <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">series. With nominate Shannon wavelet operational matrices of integration, <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">the solutions are approximated in the form of convergent series with easily <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">computable terms. Also, by applying collocation points the exact solutions <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">of fractional di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erential equations can be achieved by well-known series solutions. Illustrative examples are presented to demonstrate the applicability <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">and validity of the wavelet base technique. To highlight the convergence, <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">the numerical experiments are solved for di<span style="font-family: rtxr; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">erent values of bounded series <span style="font-family: NimbusRomNo9L-Regu; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;">approximation.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> https://wala.vru.ac.ir/article_6349_8c5149362e6e7f7947fd2f578df6b575.pdf