article · Mathematics
A hybrid numerical scheme has been developed to approximate linear Fredholm integral equation systems by combining Bernoulli polynomials with enhanced block-pulse functions. This formulation couples the local support and computational efficiency of block-pulse functions with the high-order approximation properties of Bernoulli polynomials. Operational matrices are employed to convert the coupled integral equations into a finite-dimensional algebraic system of expansion coefficients. Theoretical evaluations confirm the solvability, convergence, stability, and approximation error of the technique. Furthermore, investigations into the effects of polynomial degree and partition refinement show that the method achieves high accuracy using relatively small basis dimensions. When assessed against previously reported numerical techniques across several test problems, the approach exhibits superior computational effectiveness and precision.
Systems of integral equations are fundamental across computational science and engineering, but finding exact mathematical solutions is frequently intractable. Developing numerical schemes that require smaller basis dimensions while preserving high precision helps reduce computational overhead, enabling faster and more reliable solutions for complex scientific problems.
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This paper proposes a novel mixed numerical scheme for approximating linear Fredholm integral equation systems (LFISs) using a combination of Bernoulli polynomials (BPs) and enhanced block-pulse functions (EBPFs). This suggested representation makes use of both the local support nature and computation efficiency of the (EBPFs) as well as the high-order approximating nature of BPs. Using the operational matrices, the system of coupled integrals can be transformed into a finite-dimensional algebraic system (AS) of expansion coefficients. A theoretical analysis is established to investigate the solvability, convergence, stability, and approximation error of the resulting scheme. In addition, the effects of polynomial degree and partition refinement on the numerical accuracy are examined. Several test problems are considered, and the obtained results demonstrate that the proposed BEBPF approach provides highly accurate approximations while requiring relatively small basis dimensions. Comparisons with previously reported numerical techniques further illustrate their computational effectiveness and accuracy.
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DOI: 10.3390/math14173240
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