Abstract
A numerical technique for Volterra functional integral equations (VFIEs) with non-vanishing delays and fractional Bagley-Torvik equation is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are utilized to evaluate the accurate results. The findings for examples figs and tables show that the technique is accurate and simple to use.
Highlights
IntroductionThe following general Volterra functional integral equation (VFIE) is considered (1.1)
The following general Volterra functional integral equation (VFIE) is considered (1.1)Zt y(t) = g(t) + b(t)y(θ(t)) + k1(t, s)y(s)ds Z θ(t) t0 +k2(t, s)y(s)ds, t ∈ I := (t0, T ] = (t0, tf ], t0 y(t) = φ(t), t ∈ Iθ := [θ(t0), t0], tf = T, where 0 < q < 1 (θ(t) is the function which related to q), and k1(t, s) and k2(t, s) are assumed to be continuous functions on their respective domains D := {(t, s) : t0 ≤ s ≤ t ≤ T, t ∈ I} and Dθ := {(t, s) : θ(t0) ≤ s ≤ θ(t), t ∈ I}
Z θ(t) k2(t, s)y(s)ds, t ∈ I := (t0, T ] = (t0, tf ], t0 y(t) = φ(t), t ∈ Iθ := [θ(t0), t0], tf = T, where 0 < q < 1 (θ(t) is the function which related to q), and k1(t, s) and k2(t, s) are assumed to be continuous functions on their respective domains D := {(t, s) : t0 ≤ s ≤ t ≤ T, t ∈ I} and Dθ := {(t, s) : θ(t0) ≤ s ≤ θ(t), t ∈ I}
Summary
The following general Volterra functional integral equation (VFIE) is considered (1.1). The hp variant of the discontinuous Galerkin technique for the numerical solution of delay differential equations (DDEs) with nonlinear vanishing delays was introduced in [2]. In [5] and [6], authors utilized the Bezier curves in approximating functions. For solving differential equations (DEs) numerically, authors in [7] proposed the utilization of Bezier curves. To solve delay differential equation, the Bezier control points strategy is utilized (see [8]). We use the proposed method in [9] for solving VFIE. The outline of this sequel is as follow: In Section 2, problem statement is stated.
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