Abstract
We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very high, a fact that is illustrated by the numerical examples presented. The comparison with previous approximations computed for the included test problems emphasizes the method’s simplicity and accuracy.
Highlights
Integro-differential equations are important in both pure and applied mathematics, with multiple applications in mechanics, engineering, physics, etc
While the qualitative properties of integro-differential equations are thoroughly studied ([1,2]), leaving aside a relatively small number of exceptions, the exact solution of a nonlinear integro-differential equation of the type (1) cannot be found, and numerical solutions or approximate analytical solutions must be computed
If the problem has an exact polynomial solution, it is easy to see if PLSM has found it since the value of the minimum of the functional in this case is 0
Summary
Integro-differential equations are important in both pure and applied mathematics, with multiple applications in mechanics, engineering, physics, etc. Together with the set of conditions of (2) (if present) admits a solution We remark that this class of equations includes both Fredholm- and Volterratype equations, linear and nonlinear equations and, both integro-differential and integral equations, so it is a very general class of equations . While the qualitative properties of integro-differential equations are thoroughly studied ([1,2]), leaving aside a relatively small number of exceptions (mostly test problems, such as the ones included as examples), the exact solution of a nonlinear integro-differential equation of the type (1) cannot be found, and numerical solutions or approximate analytical solutions must be computed. Many approximation techniques have been proposed for the computation of analytic approximations of integro-differential Fredholm and Volterra equations, such as, for example, the following: Taylor expansion methods ([3,4]), Tau methods ([5,6]), the homotopy perturbation method ([7]), the Bessel polynomials method ([8]), Legendre methods ([9]), the Bernoulli matrix method ([10]), the Haar wavelet method ([11,12,13]), collocation methods ([14,15]), the Bernstein–Kantorovich operators method ([16]), Cattani’s method ([17]), the variational iteration method ([18]), the Bernstein polynomials-based projection method ([19]), the block pulse functions method ([20,21,22]), the modified decomposition method ([23]), and the differential transform method ([24])
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