In this article, a wavelet method is introduced for solving Caputo-Hadamard fractional differential equations on an arbitrary interval. The proposed method is the fractional-order generalization of sine-cosine wavelets (FGSCWs). The operational matrices of fractional-order integration are constructed for solving initial value problem as well as boundary value problem. Furthermore, numerical solution of nonlinear Caputo-Hadamard fractional differential equation is obtained with the conjunction of proposed method with quasilinearization technique. We have constructed the FGSCW operational matrix, FGSCW operational matrix of Hadamard fractional integration of arbitrary order, and FGSCW operational matrix of Hadamard fractional integration for Caputo-Hadamard fractional boundary value problems. Convergence analysis of the proposed method is investigated. Numerical procedure is given for both Caputo-Hadamard initial and boundary value problems. Illustrative examples show the reliability and efficiency of the proposed method and give solution with less error.

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

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