In this study, a collocation method based on Bernstein polynomials is developed for solution of the nonlinear ordinary differential equations with variable coefficients, under the mixed conditions. These equations are expressed as linear ordinary differential equations via quasilinearization method iteratively. By using the Bernstein collocation method, solutions of these linear equations are approximated. Combining the quasilinearization and the Bernstein collocation methods, the approximation solution of nonlinear differential equations is obtained. Moreover, some numerical solutions are given to illustrate the accuracy and implementation of the method.

Abstract:In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated.

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In this paper, the largest set in the literature of space, time and space-time conformable fractional Phi-4 equations is found by utilizing an analytical method based upon the Jacobi elliptic functions. These solutions are obtained in a general form including trigonometric, rational, complex and hyperbolic functions. Some problems are presented to illustrate the practical application of the proposed method and some of the solutions are also demonstrated with the two-dimensional and three-dimensional graphics.

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The main purpose of this study was to present an approximation method based on the Laguerre polynomials to obtain the solutions of the fractional linear Fredholm integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivative of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed. ABSTRAKTujuan utama kajian ini adalah untuk mengemukakan kaedah penghampiran berdasarkan polinomial Laguerre untuk mendapatkan penyelesaian pecahan linear persamaan pembezaan-kamiran Fredholm. Kaedah ini menjelmakan persamaan pembezaan-kamiran ke sistem persamaan aljabar linear dengan menggunakan titik-titik kolokasi. Di samping itu, hubungan matriks untuk terbitan pecahan Caputo polinomial Laguerre juga diperoleh. Selain itu, beberapa contoh dibentangkan untuk menggambarkan ketepatan kaedah dan hasilnya dibincangkan.Kata kunci: Persamaan pembezaan-kamiran Fredholm; persamaan pembezaan-kamiran pecahan; polinomial Laguerre

In this study, a collocation method, one of the type of projection methods based on the generalized Bernstein polynomials, is developed for the solution of high-order linear Fredholm-Volterra integro-differential equations containing derivatives of unknown function in the integral part. The method is valid for the mixed conditions. The convergence analysis and error bounds of the method are also given. Besides, six examples are presented to demonstrate the applicability and validity of the method. ARTICLE HISTORY