Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

Abstract

The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo-Hadamard fractional differential equations on a semi-infinite domain.Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo-Hadamard fractional delay differential equations. Findings:We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration, and Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo-Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo-Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo-Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present

Solution Method for Systems of Nonlinear Fractional Differential Equations Using Third Kind Chebyshev Wavelets

“…The other benefits can be their orthogonality, compact support, and simultaneous representation of data in several resolutions. A wide variety of FDEs have been solved using numerous wavelet basis functions [16][17][18][19][20][21][22][23][24][25].…”