In this study, we addressed the initial value problem in the context of the $ (k, \rho, \phi) $-Caputo fractional operators. The existence and uniqueness results were established via the fixed-point theory of Banach's and Krasnoselskii's types. A comprehensive qualitative analysis was conducted, including the investigation of Ulam-Hyers-Mittag-Leffler (UHML) stability. A modified numerical scheme, based on a two-step Lagrange interpolation polynomial, was developed for the $ (k, \rho, \phi) $-Caputo fractional derivative operator. A convergence analysis of the piecewise Lagrange interpolation method is presented, establishing the theoretical order of convergence. The applicability and effectiveness of the proposed method were demonstrated through selected real-world problems. Furthermore, numerical results were compared with those obtained from various fractional derivative operators and with observed data. The accuracy and performance of the analytic and proposed methods were quantitatively assessed applying $ L^{2} $-norm error, mean square error (MSE), and root mean square error (RMSE).
Citation: Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon. Numerical solutions and convergence analysis of $ (k, \rho, \phi) $-Caputo fractional differential equations with applied science applications[J]. AIMS Mathematics, 2026, 11(10): 33571-33616. doi: 10.3934/math.20261313
In this study, we addressed the initial value problem in the context of the $ (k, \rho, \phi) $-Caputo fractional operators. The existence and uniqueness results were established via the fixed-point theory of Banach's and Krasnoselskii's types. A comprehensive qualitative analysis was conducted, including the investigation of Ulam-Hyers-Mittag-Leffler (UHML) stability. A modified numerical scheme, based on a two-step Lagrange interpolation polynomial, was developed for the $ (k, \rho, \phi) $-Caputo fractional derivative operator. A convergence analysis of the piecewise Lagrange interpolation method is presented, establishing the theoretical order of convergence. The applicability and effectiveness of the proposed method were demonstrated through selected real-world problems. Furthermore, numerical results were compared with those obtained from various fractional derivative operators and with observed data. The accuracy and performance of the analytic and proposed methods were quantitatively assessed applying $ L^{2} $-norm error, mean square error (MSE), and root mean square error (RMSE).
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Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon. Numerical solutions and convergence analysis of $ (k, \rho, \phi) $-Caputo fractional differential equations with applied science applications[J]. AIMS Mathematics, 2026, 11(10): 33571-33616. doi: 10.3934/math.20261313
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