Positive Baskakov–Schurer–Stancu approximation for local truncated m -fractional source+ problems
Year: 2026
Type: article
Abstract: We propose a positive approximation scheme for source terms arising in local truncated M -fractional source problems. Here, the term ‘truncated M -fractional’ refers to a local weighted derivative: for differentiable functions, it reduces to K x 1 − ν d / d x . Thus, the benchmark considered is a scalar local initial-value problem rather than a nonlocal evolution equation with memory. The scheme combines Baskakov–Schurer–Stancu basis functions with a normalized local average induced by the truncated M -fractional integral. This construction preserves positivity, yields explicitly computable moments, and incorporates a fractional-type local averaging parameter into the operator nodes. The resulting explicit source approximation can be inserted directly into local fractional-type initial-value problems without solving an additional nonlinear algebraic system. It is particularly useful when source, density, heat-input, or wave-amplitude profiles must remain nonnegative during numerical approximation. For the resulting operators, we compute the first and second algebraic moments and the second central moment, establish weighted Korovkin convergence on the half-line, and derive direct estimates using the ordinary modulus of continuity and Lipschitz classes. We also apply the operators to an inhomogeneous problem in which the truncated M -fractional derivative acts as a local weighted derivative. The induced approximate solutions are shown to converge uniformly to the exact solution on compact intervals. A dimensionless thermal-source benchmark supports the theoretical findings: for the polynomial source considered, the grid-based maximum error decreases from 0.428002 at n = 5 to 0.042014 at n = 80 . Additional benchmark and sensitivity studies clarify the roles of α , ρ , and ϑ . The results provide a transparent bridge between Baskakov-type positive approximation and local truncated M -fractional source models, without portraying the scalar benchmark as a full heat-conduction or nonlocal memory model.
Authors: Ali Olgun, Oğuz Yağcı
Institution: Kırıkkale University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,880
OpenAlex ID: W7204249729