Locally One-Dimensional Scheme for a Multidimensional Fractional-Order Heat Equation with Conditions of the Third Kind in an Arbitrary Domain

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/f6557-1323-1446-g Locally One-Dimensional Scheme for a Multidimensional Fractional-Order Heat Equation with Conditions of the Third Kind in an Arbitrary Domain Beshtokova, Z. V. , Beshtokov, M. Kh. , Shkhanukov-Lafishev, M. Kh. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1. Abstract: The multidimensional fractional-order heat equation with boundary conditions of the third kind in a domain with a complex shape is studied. Instead of the original differential equation we consider a modified fractional order heat equation with regularization parameter \(\varepsilon>0\). The finite difference method is used for approximate solution of the modified problem. A local one-dimensional difference scheme of A. A. Samarsky with approximation order \(O(\|h\|^2+\tau)\) is constructed. The essence of this scheme is as follows. We reduce the transition from layer to layer to the sequential solution of one-dimensional problems in each of the coordinate directions. Using the maximum principle, we obtain an a priori estimate in the uniform metric in the norm \(C\). Moreover, we prove the stability of the locally uniform difference scheme and the uniform convergence of the solution of the proposed difference scheme to the solution of the original problem for any values \(0<\alpha<1\). A particular choice of the regularization parameter \(\varepsilon\) can significantly affect the convergence rate of the local-uniform difference scheme and the quality of its solution. In this manuscript we give detailed analysis of the choice of optimal values of \(\varepsilon\) such that the rate of uniform convergence of the solution of the proposed difference scheme to the solution of the original problem will be determined in the best possible way. Keywords: heat equation, fractional order equation, the~Gerasimov--Caputo fractional derivative, boundary value problems, locally one-dimensional scheme, maximum principle, a priori estimate, stability and convergence. Language: Russian Download the full text Download JATS XML For citation: Beshtokova, Z. V., Beshtokov, M. Kh. and Shkhanukov-Lafishev, M. Kh. Locally One-Dimensional Scheme for a Multidimensional Fractional-Order Heat Equation with Conditions of the Third Kind in an Arbitrary Domain, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 37-61(in Russian). DOI 10.46698/f6557-1323-1446-g + References 1. Lafisheva, M. M. and Shkhanukov-Lafishev, M. Kh. 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