Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

YÜZBAŞI Ş., YILDIRIM G.

INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS, cilt.17, sa.10, 2020 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 17 Sayı: 10
  • Basım Tarihi: 2020
  • Doi Numarası: 10.1142/s0219876220500115
  • Dergi Adı: INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, Computer & Applied Sciences, Metadex, zbMATH, Civil Engineering Abstracts
  • Anahtar Kelimeler: Collocation method, Legendre polynomials, nonlinear equations, residual correction method, Riccati type differential equations, HOMOTOPY PERTURBATION METHOD, DIFFERENTIAL-EQUATION, NUMERICAL-SOLUTION, HYBRID LEGENDRE, BLOCK-PULSE, POLYNOMIALS, CHEBYSHEV, KIND
  • Akdeniz Üniversitesi Adresli: Evet

Özet

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.