Dirac’s formalism combined with complex Fourier operational matrices to solve initial and boundary value problems

PurposeThe purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.Design/methodology/approachThe author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.FindingsThe author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.Originality/valueMany scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

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Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

Abstract and Applied Analysis ◽

10.1155/2015/672703 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

W. M. Abd-Elhameed

Keyword(s):

Boundary Value Problems ◽

Jacobi Polynomials ◽

Boundary Value ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Third Order ◽

Generalized Jacobi Polynomials ◽

Nonlinear Algebraic Equations ◽

Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

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Direct Computation of Operational Matrices for Polynomial Bases

Mathematical Problems in Engineering ◽

10.1155/2010/139198 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12 ◽

Cited By ~ 6

Author(s):

Osvaldo Guimarães ◽

José Roberto C. Piqueira ◽

Marcio Lobo Netto

Keyword(s):

Hilbert Space ◽

Numerical Methods ◽

Boundary Value Problems ◽

Boundary Value ◽

Operational Matrices ◽

Direct Computation ◽

Linear Operation ◽

Matrix Operations ◽

Polynomial Bases ◽

Reference Matrix

Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

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