Abstract:
Differential equations are frequently used to describe continuous-time dynamical
system. Not all differential equations can be solved analytically, so several
techniques have been developed to find approximate solutions. However, these
methods are restricted to certain types of differential equations. Non-homogenous
ordinary linear differential equations with variable coefficients are the very general
class of ordinary linear differential equations. The intention of this work is to
present an integral collocation method by using shifted Chebyshev polynomials of
first kind to find an approximate solution of a linear differential equation with
variable coefficients. The method starts with writing the highest derivative of the
unknown function in a given differential equation as truncated shifted Chebyshev
series in analytical form. Then, lower order derivatives and the unknown function
are obtained by means of successive integrations and substituted in the given
differential equation. Thereby, the equation reduces to an algebraic equation with
unknown coefficients. Chebyshev nodes, given initial and or boundary condition
values are used as shrewd collocation points to determine the Chebyshev unknown
coefficients and integral constants. Three numerical examples with different initial
and or boundary conditions are discussed to show the efficacy of the proposed
method. An advantage of the method over conventional numerical method is that
the solution is in polynomial form. Thus, solution obtained by this method can be
used to interpolate other functional value in the domain. Also, it illustrated through
the numerical examples presented that when the degree of the approximating
polynomial increases, the approximate solution converges to exact solution.
