Bernoulli Wavelets Method For Solving Three Kinds Of System Of Fractional Calculus Equations

With the development of science and technology, more and more practical issues which are contained in different disciplines such as engineering and physics are fractional order systems after the establishment of the mathematical model, and most of these systems need to be described by the fractional calculus equations, so solving the numerical solutions of fractional calculus equations is the premise to deal with these systems. Fractional calculus as a hot topic has attracted considerable interest for decades. As a consequence, many numerical methods have been developed to provide approximated solutions for the fractional calculus equations. Such as Adomian decomposition method, Laplace transforms, collocation method, and Taylor expansion method. However, the researches of using wavelets method to solve systems of fractional calculus equations are relatively few. At present, the most used orthogonal wavelets are Haar wavelets, Legendre wavelets, Chebyshev wavelets and so on. Here, in this paper, a new wavelets is proposed, it is the Bernoulli wavelets. Meanwhile, a new numerical method for fractional calculus equations is presented, which is based upon Bernoulli wavelets approximation. By combining Bernoulli wavelets with its fractional integral operator matrix, the problem can be transformed into a system of linear or nonlinear algebraic equations, which can be solved easily via Matlab software.Firstly, we briefly introduced the development and application of wavelets analysis and fractional calculus equations, and then proposed the research background and significance of this artical.Secondly, we simply outlined some basic knowledge, gave three definitions and the related links of fractional calculus, also introduced the formation of wavelet, Bernoulli wavelet and Block Pulse Function(BPFs) and its associated properties.Finally, a detailed analysis of Bernoulli wavelets approach for fractional calculus equations is presented. Meanwhile, the error analysis and illustrative examples are included to reveal the effectiveness of the technique. The method is based on Bernoulli wavelets approximation, by using the properties of Bernoulli wavelets and BPFs, derived the fractional integral operator matrix of Bernoulli wavelets, and then combined Bernoulli wavelets and its fractional integral operator matrix to transform the studied fractional calculus equations into a liner or nonlinear system of algebraic equations, which can be solved easily via Matlab software.