Abstract
We develop the multiwavelet Galerkin method to solve the Volterra–Fredholm integral equations. To this end, we represent the Volterra and Fredholm operators in multiwavelet bases. Then, we reduce the problem to a linear or nonlinear system of algebraic equations. The interesting results arise in the linear type where thresholding is employed to decrease the nonzero entries of the coefficient matrix, and thus, this leads to reduction in computational efforts. The convergence analysis is investigated, and numerical experiments guarantee it. To show the applicability of the method, we compare it with other methods and it can be shown that our results are better than others.
Highlights
A mathematical model of the spatiotemporal development of an epidemic yields the following Volterra–Fredholm integral equation (VFIE):x u(x) f(x) + k1(x, s, u(s))ds (1)+ k2(x, s, u(s))ds, x ∈ Ω ≔ [0, 1], where the given functions f: Ω ⟶ R and k1: S × R ⟶ R with S ≔ {(x, s): x, s ∈ Ω} are assumed to be continuous functions
E parabolic boundary value problems lead to these types of integral equations and widely arise from various physical and biological models. e VFIE appears in the literature in mixed form as x1 u(x) f(x) + k(t, s)u(s)ds dt, (2)
We focus on a paper that uses the multiwavelet Galerkin method to solve linear mixed VFIE as mentioned in [9]
Summary
A mathematical model of the spatiotemporal development of an epidemic yields the following Volterra–Fredholm integral equation (VFIE):. We can mention collocation method [1], projection method [2], spline collocation method [3], wavelet collocation method [4], Adomian decomposition method [5], and so on [6,7,8] Among these studies, we focus on a paper that uses the multiwavelet Galerkin method to solve linear mixed VFIE as mentioned in [9]. In [9], the wavelet transform matrix and the operational matrix of integration are utilized to reduce the problem of linear mixed VFIE to a sparse linear system of algebraic equations. Journal of Mathematics the multiwavelet Galerkin method, both types reduce to the system of the linear and nonlinear algebraic equations, respectively.
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