A Second-Order Numerical Approximation for Volterra-Fredholm Integro-Differential Equations with Boundary Layer and an Integral Boundary Condition
A Second-Order Numerical Approximation
Keywords:
Singular perturbation; Integro-differential equation; Finite difference methods; Piecewise uniform mesh; Uniform convergentAbstract
This study introduces a novel second-order computational technique to
effectively tackle Volterra Fredholm integro-differential equations, which
are characterized by integral conditions and boundary layers. Initially,
some analytical properties of the solution are given. Then, the approach
involves implementing a finite difference scheme on the piece-wise uniform
mesh (Shishkin-type mesh). It integrates a composite trapezoidal formula for
the integral component and utilizes interpolating quadrature rules and
linear exponential basis functions for the differential part. The analysis
of the method demonstrates that both the numerical scheme and its
convergence rate exhibit second-order accuracy, ensuring uniform convergence
with respect to the small parameter in the discrete maximum norm. Finally,
two test examples are given.
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