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In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order α$$ \alpha $$, and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. Design/methodology/approach: The authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique. Findings: The authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions. Originality/value: Since psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors' knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we develop the generalized fractional order Chebyshev wavelets (GFCWs) from generalized fractional order of Chebyshev polynomials. The operational matrices for the presented wavelets are constructed and derived. We also proposed a technique by utilizing the GFCWs, the method of steps and quasilinearization technique for solving nonlinear fractional delay-type differential equations. According to the development, the method of step is used to transform the fractional nonlinear delay-type differential equation to a fractional nonlinear non-delay differential equation, and then apply the quasilinearization technique to discretize the obtained nonlinear equation. The GFCW method is utilized in each iteration of quasilinearization method for the improvement of solution. We perform the error analysis for the proposed technique. Procedure of implementation for the present method is also provided. Numerical simulation of some examples will be presented to demonstrate the benefits of computing with the present technique over existing methods in literature. [ABSTRACT FROM AUTHOR]

In this article, we propose a method for the solution of the generalized Burger-Fisher equation. The method is developed using CAS wavelets in conjunction with quasi-linearization technique. The operational matrices for the CAS wavelets are derived and constructed. Error analysis and procedure of implementation of the method are provided. We compare the results produce by present method with some well known results and show that the present method is more accurate, efficient, and applicable. [ABSTRACT FROM AUTHOR]

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique. [ABSTRACT FROM AUTHOR]

A numerical method is proposed by wavelet-Galerkin and quasilinearization approach for nonlinear boundary value problems. Quasilinearization technique is applied to linearize the nonlinear differential equation and then wavelet-Galerkin method is implemented to linearized differential equations. In each iteration of quasilinearization technique, solution is updated by wavelet-Galerkin method. In order to demonstrate the applicability of proposed method, we consider the various nonlinear boundary value problems. [ABSTRACT FROM AUTHOR]

We showed that solutions by the Haar wavelet-quasilinearization technique for the two problems, namely, (i) temperature distribution equation in lumped system of combined convection-radiation in a slab made of materials with variable thermal conductivity and (ii) cooling of a lumped system by combined convection and radiation are strongly reliable and also more accurate than the other numericalmethods and are in good agreement with exact solution. According to theHaar wavelet-quasilinearization technique, we convert the nonlinear heat transfer equation to linear discretized equation with the help of quasilinearization technique and apply the Haar wavelet method at each iteration of quasilinearization technique to get the solution. The main aim of present work is to show the reliability of the Haar wavelet-quasilinearization technique for heat transfer equations. [ABSTRACT FROM AUTHOR]

Abstract: In this article, numerical solutions of nonlinear ordinary differential equations of fractional order by the Haar wavelet and quasilinearization are discussed. Quasilinearization technique is used to linearize the nonlinear fractional ordinary differential equation and then the Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Haar wavelet method. The results are compared with the results obtained by the other technique and with exact solution. Several initial and boundary value problems are solved to show the applicability of the Haar wavelet method with quasilinearization technique. [Copyright &y& Elsevier]

In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. The method simplifies the existing CAS wavelet method and does not require conventional operational matrices of integration for certain cases. Quasilinearization technique is used to transform non-linear fractional differential equations to linear equations and then Green-CAS method is applied. Furthermore, the proposed method has also been analyzed for convergence, particularly in the context of error analysis. Sufficient conditions for the existence of unique solutions are established for the boundary value problem under consideration. Moreover, to elaborate the effectiveness and accuracy of the proposed method, results of essential numerical applications have also been documented in graphical as well as tabular form. [ABSTRACT FROM AUTHOR]