Abstract
In this paper, we present a computational method to solve fractional Klein-Gordon equation (FKGE). The proposed technique is the grouping of orthogonal polynomial matrices and collocation method. The benefit of the computational method is that it reduces the FKGE into a system of algebraic equations which makes the problem straightforward and easy to solve. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. The main solution behaviours of these equations are due to fractional orders which are explained graphically.Numerical results obtained by proposed computational method are also compared with the exact solution. The results obtained by suggested technique reveals that the method is very useful for solving FKGE.
Highlights
The standard Klein-Gordon equation (KGE) is written as ∂2v ∂2v∂t2 − ∂x2 + v = h (x, t), x ≥ 0, t ≥ 0 (1)where v indicates an unknown function in variables x and t, and h(x, t) stands for the source term
First the unknown function and their derivatives are approximated by taking finite dimensional approximations. By using these approximations along with operational matrices of differentiations and integrations in the fractional Klein-Gordon equation (FKGE), we obtain a system of equations
By making use of the value of C in Equation (13), we can obtain an approximate solution for FLGE
Summary
Where v indicates an unknown function in variables x and t, and h(x, t) stands for the source term. We have used Chebyshev polynomials as a basis function for the construction of operational matrices of differentiations and integrations. First the unknown function and their derivatives are approximated by taking finite dimensional approximations. By using these approximations along with operational matrices of differentiations and integrations in the FKGE, we obtain a system of equations. ., Hn]T, is Chebyshev vector and we consider β > 0, where, D(γ )is the operational matrix of differentiation of order γ and is given by Equation (11). Where, D(β) = s i, j , is (n + 1) × (n + 1) matrix of differentiation of non-integer order β and its entries are given by h(x, t) = θnT(x)Eθn(t). By making use of the value of C in Equation (13), we can obtain an approximate solution for FLGE
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