Abstract
This article studies the Ulam–Hyers stability of the second-order convergent finite difference scheme for the first- and second-order non-homogeneous linear differential equations \(x'(t)-bx(t)=f(t)\) and \(x''(t)+\alpha x'(t)+\beta x(t)=g(t),\) on the interval \(I=[a,\infty ),\) respectively, where \(f,g:I\rightarrow {\mathbb {R}}\) are given functions and \(a,b,\alpha ,\beta \in {\mathbb {R}}.\) After converting the finite difference scheme to its equivalent linear recurrence relation and by using the Ulam–Hyers stability results for the linear recurrence relation, we establish the Ulam–Hyers stability for the finite difference scheme. Further, as per the location of the roots of the characteristic polynomial of the equivalent recurrence relation, the minimum Ulam–Hyers constant is determined. To illustrate the utility of the obtained result, we apply our result to the perturbed second-order nonlinear difference equation and present a suitable example at the end to support the obtained result.
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Acknowledgements
The second author expresses his gratitude to Indian Institute of Technology Guwahati, India for providing him a research fellowship to carry out research towards his PhD. Both authors express their gratefulness to the anonymous esteemed Reviewer for his time and effort in going through the manuscript and for giving suggestions for improvement. Associate Editor Prof. Ioan Rasa is profusely thanked for handling the manuscript and for giving an opportunity to revise.
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The second author received junior research fellowship from Indian Institute of Technology Guwahati for the period 2019–2020 and senior research fellowship for the period 2021-present.
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Conceptualization: Matap Shankar; Methodology: Matap Shankar; Formal analysis and investigation: Matap Shankar, Swaroop Nandan Bora; Writing–original draft preparation: Matap Shankar, Swaroop Nandan Bora; Writing–review and editing: Matap Shankar, Swaroop Nandan Bora.
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Bora, S.N., Shankar, M. Ulam–Hyers Stability of Second-Order Convergent Finite Difference Scheme for First- and Second-Order Nonhomogeneous Linear Differential Equations with Constant Coefficients. Results Math 78, 17 (2023). https://doi.org/10.1007/s00025-022-01791-5
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DOI: https://doi.org/10.1007/s00025-022-01791-5