Abstract
In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method.
Highlights
In recent years, many studies have been developed about fractional partial differential equations
Fibonacci polynomials In this study, we introduce Fibonacci collocation method based on matrix relations which has been used to find the approximate solutions of some classes of the differential equations such as integro-differential equations, differential-difference equations, Fredholm integro differential-difference equations, Pantograph-type functional differential equations and linear Volterra integro differential equations
Fibonacci collocation method is presented for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions (Kurt et al 2013a, b) and this method is used to solve both the linear Fredholm integro-differential-difference equations (Kurt et al 2013a, b) and high-order Pantograph-type functional differential equations (Kurt Bahşı et al 2015)
Summary
Many studies have been developed about fractional partial differential equations. Fibonacci polynomials In this study, we introduce Fibonacci collocation method based on matrix relations which has been used to find the approximate solutions of some classes of the differential equations such as integro-differential equations, differential-difference equations, Fredholm integro differential-difference equations, Pantograph-type functional differential equations and linear Volterra integro differential equations. Fibonacci collocation method is presented for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions (Kurt et al 2013a, b) and this method is used to solve both the linear Fredholm integro-differential-difference equations (Kurt et al 2013a, b) and high-order Pantograph-type functional differential equations (Kurt Bahşı et al 2015). On the other hand this method is applied for the linear Volterra integro differential equations (Kurt Bahşı and Yalçınbaş 2016a, b).
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