Kronecker operational matrices for fractional calculus and some applications
Introduction
One of the principle reasons is that matrices arise naturally in communication systems, economic planning, statistics, control theory and other fields of pure and applied mathematics [2], [4], [7]. Another is because an m × n matrix can be valued as representing a linear map from an n-dimensional vector space to an m-dimensional vector space, and conversely, all such maps can be represented as m × n matrices. The operational matrices for integration and differentiation are studied by many authors [1], [3], [6], [10]. For example, Mouroutsos and Sparis [6] solved the control problem using Taylor series operational matrices for integration, Chen et al. [3] solved the distributed systems by using Walsh operational matrices and Wang [10] introduced the inversions of rational and irrational transfer functions by using the generalized block pulse operational matrices for differentiation and integration. The Kronecker convolution product is an interesting area of current research and in fact plays an important role in applications, for example, Sumita [9], established the matrix Laguerre transform to calculate matrix convolutions and evaluated a matrix renewal function.
In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann–Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and numerical solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.
As usual, the notations A{−1}(t) and det A(t) are the inverse and determinant of matrix function A(t), respectively, with respect to convolution. The notations AT(t) and Vec A(t) are transpose and vector-operator of matrix function A(t), respectively. The term “Vec A(t)” will be used to transform a matrix A(t) into a vector by stacking its column one underneath.The notations δ(t) and Qn(t) = Inδ(t) are the Dirac delta function and Dirac identity matrix, respectively, where In is the identity scalar matrix of order n × n. Finally, the notations A(t) ⊛ B(t) and A(t) ∗ B(t) are the convolution and Kronecker convolution products, respectively. Finally, A ⊗ B stands for the Kronecker product ; A ⊗ B = [aijB]ij.
Section snippets
Taylor series operational matrix for integration
A function f(t) that is analytic function at the neighborhood of the point t0 can be expanded in the following formula:where and φn(t) = tn.
To obtain an approximate expression of the analytic function f(t) we may truncate the series (1) up to the (r + 1)th term:By defining the coefficient vector γT as:and the power series basis vector φT(t) as:Eq. (2) can written in the following compact form:The basis
Convolution product and Riemann–Liouville fractional integral of matrices
Definition 1 Let A(t) = [fij(t)] and B(t) = [gij(t)] be m × m absolutely integrable matrices on [0, b). The convolution and Kronecker convolution product s are matrix functions defined by Convolution product: Kronecker convolution productHere, fij(t) ∗ B(t) is the ijth submatrix of order m × m and A(t) ⊛ B(t) is of order m2 × m2.
Definition 2
Let and B(t) = [gij(t)] be m × m absolutely integrable matrices on [0, b). The
Solution of the state–space equations and optimal control problem using Taylor series and Kronecker product
The state–space equation is given bywhere , analytic and A, B are known constant matrices. The input vector u(t) may be expanded in Taylor series as follows:where H is a known constant matrix.
Similarly, the state vector x(t) may also be expanded in Taylor series as follows:
Acknowledgements
The present research has been partially supported by University Putra Malaysia (UPM) under the grant IRPA09-02-04-0259-EA001.
References (10)
- et al.
Numerical solution of integral equations systems of second kind by Block Pulse Functions
Appl. Math. Comput.
(2005) - et al.
Taylor Series approach to system identification, analysis and optimal control
J. Frankin Inst.
(1985) - Al Zhour Zeyad, Kilicman Adem, Some applications of the convolution and Kronecker products of matrices, in: Proceeding...
- et al.
Optimal Sampled-Data Control Systems
(1995) - et al.
Walsh operational matrices for fractional calculus and their applications to distributed systems
J. Frankin Inst.
(1977)
Cited by (167)
Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment
2024, Applied Soft ComputingA new wavelet method for fractional integro-differential equations with ψ-Caputo fractional derivative
2024, Mathematics and Computers in SimulationSolving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics
2024, Ain Shams Engineering JournalShifted fractional order Gegenbauer wavelets method for solving electrical circuits model of fractional order
2023, Ain Shams Engineering JournalAn application of Genocchi wavelets for solving the fractional Rosenau-Hyman equation<sup>☆</sup>
2021, Alexandria Engineering JournalNumerical Laplace inverse based on operational matrices for fractional differential equations
2024, International Journal of Dynamics and Control