An exponential method to solve linear Fredholm-Volterra integro-differential equations and residual improvement

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YÜZBAŞI Ş.

TURKISH JOURNAL OF MATHEMATICS, cilt.42, sa.5, ss.2546-2562, 2018 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 42 Sayı: 5
  • Basım Tarihi: 2018
  • Doi Numarası: 10.3906/mat-1707-66
  • Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Sayfa Sayıları: ss.2546-2562
  • Anahtar Kelimeler: Collocation method, exponential polynomials, exponential solutions, Fredholm-Volterra integro-differential equations, initial boundary conditions, residual improvement, HOMOTOPY PERTURBATION METHOD, HAAR FUNCTIONS METHOD, NUMERICAL-SOLUTION, INTEGRAL-EQUATIONS, COLLOCATION METHOD, TAU METHOD, SYSTEM
  • Akdeniz Üniversitesi Adresli: Evet

Özet

In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm-Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods.