A singularity-inferred operator-consistent fractional PINN for one- and two-dimensional third-kind Caputo–Hadamard auto-convolution Volterra equations with weakly singular kernels
Results in Engineering, cilt.32, 2026 (ESCI, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 32
- Basım Tarihi: 2026
- Doi Numarası: 10.1016/j.rineng.2026.113375
- Dergi Adı: Results in Engineering
- Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
- Anahtar Kelimeler: Auto-convolution, Caputo–Hadamard derivative, Fractional physics-informed neural network, Gauss–Jacobi quadrature, Third-kind Volterra equation, Weakly singular kernel
- İstanbul Gelişim Üniversitesi Adresli: Evet
Özet
We propose a singularity-inferred operator-consistent fractional physics-informed neural network for one- and two-dimensional third-kind Caputo–Hadamard auto-convolution Volterra equations with weakly singular kernels. The logarithmic change of variable transforms the Caputo–Hadamard derivative into a Caputo derivative, and a derivative-free Gauss–Jacobi rule is used for its numerical evaluation. The trial function contains a trainable endpoint coordinate, a polynomial term, cubic polyharmonic-spline terms, and a smooth correction. The same endpoint coordinate is used in the fitted quadrature rules. The numerical examples use an independent integral residual and higher-order evaluations of the Caputo and auto-convolution operators. In the controlled one-dimensional comparison, the proposed approximation gives the smallest mean validation error, and the two-dimensional example recovers the predicted endpoint scale.