scholarly journals Direct method for solution variational problems by using Hermite polynomials

2021 ◽  
Vol 39 (6) ◽  
pp. 223-237 ◽  
Author(s):  
Ayatollah Yari ◽  
Mirkamal Mirnia
Keyword(s):  
Optimal Control ◽  
Numerical Approximation ◽  
Optimal Control Problems ◽  
Direct Method ◽  
Hermite Polynomials ◽  
Variational Problems ◽  
Computational Method ◽  
Control Problems ◽  
Operational Matrices

‎In this approach‎, ‎one‎ computational method is presented for numerical approximation of variational problems‎. ‎This method with variable ‎coeffici‎ents is based on Hermite polynomials‎. ‎The properties of Hermite polynomials with the operational matrices of derivative and integration are used to reduce optimal control problems to the solution of linear algebraic equations‎. ‎Illustrative examples are included to demonstrate the validity and applicability of the technique‎.  

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

2022 ◽  
pp. 107754632110593
Author(s):  
Mohammad Hossein Heydari ◽  
Mohsen Razzaghi ◽  
Zakieh Avazzadeh
Keyword(s):  
Optimal Control ◽  
Fractional Integral ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Integration Method ◽  
Direct Method ◽  
Fractional Integration ◽  
Basis Functions ◽  
Control Problems ◽  
Bernoulli Functions

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

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