Research Article
An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation
Myong Hyok Sin*
,
Ryu Song Jo
Issue:
Volume 12, Issue 1, March 2026
Pages:
1-11
Received:
19 October 2025
Accepted:
4 November 2025
Published:
7 January 2026
Abstract: In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.
Abstract: In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is...
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Research Article
The Ordered Rectangular Band of Reflexive Generalized Inverses of a Matrix
Sathi P H*,
P. G. Romeo
Issue:
Volume 12, Issue 1, March 2026
Pages:
12-17
Received:
23 March 2026
Accepted:
3 April 2026
Published:
29 May 2026
DOI:
10.11648/j.ml.20261201.12
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Abstract: This article studies the algebraic structure of the set of all reflexive generalized inverses of a real matrix using a sandwich-type binary operation and examines the compatibility of certain matrix order relations with this operation. The concept of generalized inverses arises when dealing with singular or rectangular matrices, where a standard inverse does not exist. The study shows that equipped with a sandwich operation,the entire set of reflexive generalized inverses of a matrix forms the structure of a particular type of semigroup, known as a rectangular band. Further, several algebraic properties of this semigroup are investigated in detail. In particular, the compatibility of certain well-known matrix order relations, namely Sussman's order and Mitsch's order, with the sandwich operation is examined. It is shown that these order relations are preserved under the defined operation, which enables the semigroup of reflexive generalized inverses to be viewed naturally as an ordered matrix semigroup. The results obtained in this study contribute to a deeper understanding of the relationship between generalized inverse theory, semigroup structures, and matrix partial orders, thereby providing a useful framework for further research on algebraic and order-theoretic properties of generalized inverses and their applications in diverse fields.
Abstract: This article studies the algebraic structure of the set of all reflexive generalized inverses of a real matrix using a sandwich-type binary operation and examines the compatibility of certain matrix order relations with this operation. The concept of generalized inverses arises when dealing with singular or rectangular matrices, where a standard in...
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