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Observability Inequalities for Hermite Bi-cubic Orthogonal Spline Collocation Methods of 2-D Integro-differential Equations in the Square Domains

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Abstract

We will consider the problem of observability inequality of the Hermite bi-cubic orthogonal spline collocation space semi-discretizations of the 2-D integro-differential equations in the square domains. We prove the uniform (with respect to mesh-size) observability inequality in a subspace of solutions generated by the low frequencies of the negative part, and the middle frequencies of the positive part. Our method uses previously known uniform observability inequalities in the 1-d case and a dyadic spectral time decomposition. Some numerical results are presented to illustrate our theoretical analysis.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China, contract grant numbers 11271123, 11671131.

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  1. Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Da Xu

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This work was supported in part by the National Natural Science Foundation of China, contract grant numbers 11671131, 11271123.

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Xu, D. Observability Inequalities for Hermite Bi-cubic Orthogonal Spline Collocation Methods of 2-D Integro-differential Equations in the Square Domains. Appl Math Optim 84, 1341–1372 (2021). https://doi.org/10.1007/s00245-020-09680-5

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