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Solvability and approximation of nonlinear functional mixed Volterra-Fredholm equation in Banach space. (English) Zbl 1515.65332

Summary: This study probes into the existence of a unique solution and the numerical approximation of a nonlinear functional Volterra-Fredholm integral equations of the mixed type and second kind. Based on the Lipschitz constants of the functional and kernel, a Bielecki’s norm is defined and used to modify a distance inequality on a constructed self-map. The map is shown to be contractive, thereby establishing solvability. The problem is then approximated by collocating at discrete points and use of a composite multidimensional numerical quadrature approximation. A new Grönwall-type inequality is proposed, and used, to prove the second order of convergence of the numerical scheme. Numerical experiments are provided to verify the theoretical results.

MSC:

65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
65D32 Numerical quadrature and cubature formulas

Software:

BOUT++
PDFBibTeX XMLCite
Full Text: DOI Link

References:

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