Algorithm for Computing Exact Solution of the First Order Linear Differential System

Khalil, Amal; Shehata, Mohammed

doi:10.21608/sjsci.2022.140062.1002

Algorithm for Computing Exact Solution of the First Order Linear Differential System

Document Type : Regular Articles

Authors

Amal Khalil ¹
Mohammed Shehata ²

¹ Department of mathematics, faculty of science, unverstry of sohag

² Department of Basic Science, Bilbeis Higher Institute for Engineering, Ministry of Higher Education, Egypt

10.21608/sjsci.2022.140062.1002

Abstract

In this paper, we develop an algorithm for solving nonhomogeneous first order linear differential systems by using the Jordan decomposition and convert this algorithm into a Maple procedure to find the exact solution to many-variable systems.

Differential equations play an important role in the understanding of physical sciences. Many differential equations arise from problems in physics, engineering, and other sciences, and these equations serve as mathematical models for solving numerous problems in science and engineering([1]-[11]).Numerous numerical methods exist for solving differential equations, such as Taylor, Picard, Euler, Runge-Kutta and transformation methods.

Using the Jordan decomposition method, we can simplify the exponential matrix into a product of matrices. This allows us to easily find the exact solution to the system.

After the introduction, the paper is divided into six sections. In section 2, we introduce the definition and properties of matrix exponentials. Then, in section 3, we provide an overview of Jordan decomposition for matrices. Section 4 presents our algorithm. In section 5, we obtain the Maple procedure. In section 6 we apply the algorithm to find the exact solution to some linear differential systems. Finally, we concluded our result

Keywords

Main Subjects

Pure mathematics

References

[1] Abdel-Halim H. I.H. Application to differential transformation method for solving systems of differential equations, Applied Mathematical Modelling, 2008 32, 2552–2559.

[2] Agarwal R. P & Jinhai C.. Periodic solutions for first order differential systems, Applied Mathematics Letters, 2010, 23, 337-341.

[3] Higazy. M and Sudhanshu A..Sawi transformation for system of ordinary differential equations with application, Ain Shams Engineering Journal, 2021, 12 3173–3182.

[4] Mardanov, M. J, Sharifov,Y. A, Ismayilova, K. E and Zamanova, S. A. Existence and uniqueness of solutions for the system of first-order nonlinear differential equations with three-point and integral boundary conditions, European Journal of Pure and Applied Mathematics, 2019, Vol. 12, No. 3, 756-770.

[5] Sankar P. M.; Susmita R. and Biswajit D. ;Numerical Solution of First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application, International Journal of Differential Equations, , 2016 Volume,1-14.

[6] Francesco C. and Farrin P;. Solution of the system of two coupled first-order ODEs with scond-degree polynomial right-hand sides, Math Phys Anal Geom, 2021, 24-29.

[7] Mardanov. M.J; Sharifov.Y.A, Ismayilova.K.E, Zamanova.S.A.. Existence and uniqueness of solutions for the system of first-order nonlinear differential equations with three-point and integral boundary conditions. European Journal of pure and Applied mathematics, 2019 .Vol. 12, No. 3, 756-770 .

[8] Mohamed A.R.; Abd-El-Aziz E.and Mahmoud N.S.. Numerical solution of system of first-order delay differential equations using polynomial spline functions, International Journal of Computer Mathematics, 2006 Vol. 83, No. 12925–937.

[9] Mohammed S.. Exact Solution of First Order Constraints Quadratic Optimal Control Problem, Advanced Modeling and Optimization, 2015, Volume 17 , Number 1.

[10] David L.; Steven L. and Judi M.; Linear Algabra and Its Applications, 5th Edition, 2014.

[11] Mohamed K. and Bilal C.; Elementary Proofs of the Jordan Decomposition Theorem for Niloptent Matrices, American Journal of Mathematics and Statistics, 2016 ,6(6): 238-241

[12] Sheldon A.; Linear Algebra Done Right, Third edition,Springer International Publishing., 2015.

[13] Yulei R.; Jiying W. and Yong C.; Jordan decomposition and geometric multiplicity for a class of non-symmetric Ornstein-Uhlenbeck operators, Advances in Difference Equations, 2014, 34.