Analytical and Numerical Treatment of a Mixed Volterra–Fredholm Integral Equation of the First Kind with a Hypergeometric Kernel
- 1Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
Abstract
This paper investigates a mixed Volterra–Fredholm integral equation of the first kind involving a hypergeometric spatial kernel and a continuous Volterra kernel in time. The analysis is formulated in an admissible even spatial space: the unweighted even subspace ofwhen appropriate, or the corresponding weighted even space for weakly singular cases, together with continuous time dependence and the associated supremum-in-time norm. The temporal and spatial contributions are separated at the operator level. Under appropriate temporal regularity and nondegeneracy assumptions, together with the regularity, endpoint, and integrability conditions required for the successive Abel-type inversions, the temporal first-kind equation is transformed into an equivalent Volterra equation of the second kind, while the reduced spatial equation is inverted analytically through successive Abel-type transformations. The logarithmic, Carleman, and elliptic kernels are recovered as important special cases of the hypergeometric structure. For the numerical treatment, structured Toeplitz–Hankel discretizations are considered for the logarithmic and Carleman kernels. Weighted solutions are introduced to account for their different endpoint singularities, and the computations are assessed through maximum and relative errors, discrete relative L2-errors, independent residuals, experimental convergence rates, and spectral condition numbers. The results show that the Carleman approximation has more uniform maximum-norm convergence, whereas the logarithmic error is strongly localized near the singular endpoint. Over the tested discretization range, the least-squares fitted maximum-norm rates are approximately 0.086 for the logarithmic kernel and 0.469 for the Carleman kernel. A manufactured space–time benchmark is also considered for an exponentially decaying memory kernel. The response curves illustrate the effect of the memory-decay parameter, and the trapezoidal temporal discretization exhibits approximately second-order convergence. The combined analytical and numerical results demonstrate how the spatial singularity structure and temporal memory influence the accuracy, conditioning, and evolution of the mixed integral problem.
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BibTeX
@article{jan2026analytical,
title = {{Analytical and Numerical Treatment of a Mixed Volterra–Fredholm Integral Equation of the First Kind with a Hypergeometric Kernel}},
author = {Azhar Rashad Jan},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15110806},
url = {https://www.mdpi.com/2075-1680/15/11/806},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Analytical and Numerical Treatment of a Mixed Volterra–Fredholm Integral Equation of the First Kind with a Hypergeometric Kernel
AU - Azhar Rashad Jan
PY - 2026
DA - 2026/10/29
JO - Axioms
DO - 10.3390/axioms15110806
UR - https://www.mdpi.com/2075-1680/15/11/806
AB - This paper investigates a mixed Volterra–Fredholm integral equation of the first kind involving a hypergeometric spatial kernel and a continuous Volterra kernel in time. The analysis is formulated in an admissible even spatial space: the unweighted even subspace ofwhen appropriate, or the corresponding weighted even space for weakly singular cases, together with continuous time dependence and the associated supremum-in-time norm. The temporal and spatial contributions are separated at the operator level. Under appropriate temporal regularity and nondegeneracy assumptions, together with the regularity, endpoint, and integrability conditions required for the successive Abel-type inversions, the temporal first-kind equation is transformed into an equivalent Volterra equation of the second kind, while the reduced spatial equation is inverted analytically through successive Abel-type transformations. The logarithmic, Carleman, and elliptic kernels are recovered as important special cases of the hypergeometric structure. For the numerical treatment, structured Toeplitz–Hankel discretizations are considered for the logarithmic and Carleman kernels. Weighted solutions are introduced to account for their different endpoint singularities, and the computations are assessed through maximum and relative errors, discrete relative L2-errors, independent residuals, experimental convergence rates, and spectral condition numbers. The results show that the Carleman approximation has more uniform maximum-norm convergence, whereas the logarithmic error is strongly localized near the singular endpoint. Over the tested discretization range, the least-squares fitted maximum-norm rates are approximately 0.086 for the logarithmic kernel and 0.469 for the Carleman kernel. A manufactured space–time benchmark is also considered for an exponentially decaying memory kernel. The response curves illustrate the effect of the memory-decay parameter, and the trapezoidal temporal discretization exhibits approximately second-order convergence. The combined analytical and numerical results demonstrate how the spatial singularity structure and temporal memory influence the accuracy, conditioning, and evolution of the mixed integral problem.
SN - 2075-1680
ER -