An operational matrix of integration based on Haar wavelets is established, and a procedure for applying the matrix to analyse lumped and distributed-parameters dynamic systems is formulated. The technique can be interpreted from the incremental and multiresolution viewpoint. Crude as well as accurate solutions can be obtained by changing the parameter m; in the mean time, the main features of the solution are preserved. Several nontrivial examples are included for demonstrating the fast, flexible and convenient capabilities of the new method.

Haar wavelet method for solving lumped and distributed-parameter systems

An operational matrix of integration P based on Legendre wavelets is presented. A general procedure for forming this matrix is given. Illustrative examples are included to demonstrate the validity and applicability of the matrix P.

The legendre wavelets operational matrix of integration

We have applied the concept of fractional derivatives/integrals in several specific electromagnetic problems, and have obtained promising results and ideas that demonstrate that these mathematical operators can be interesting and useful tools in electromagnetic theory. We give a brief review of the general principles, definitions, and several features of fractional derivatives/integrals, and then we review some of our ideas and findings in exploring potential applications of fractional calculus in some electromagnetic problems.

/pdf/on-the-role-of-fractional-calculus-in-electromagnetic-theory-2w91d8bjs3.pdf

On the role of fractional calculus in electromagnetic theory

Kronecker operational matrices for fractional calculus and some applications

The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.

Chebyshev series approach to system identification, analysis and optimal control

The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.

C.F. Cheng

Papers

Walsh operational matrices for fractional calculus and their application to distributed systems