This paper introduces a new method for solving linear Fredholm integral equations of the second kind by utilizing Bernoulli polynomials combined with improved block-pulse functions. The main goal is to create a numerical method that is both efficient and accurate that leverages the properties of Bernoulli polynomials for approximating the solution while utilizing the flexibility and simplicity of block-pulse functions for discretization. The improved block-pulse function enhances the representation of the solution by offering better convergence and accuracy. By combining these two mathematical tools, the method provides a robust framework for addressing Fredholm integral equations, which are commonly encountered in areas like physics, engineering, and applied mathematics, the proposed method's effectiveness is illustrated through multiple examples, highlighting its capacity to provide precise and computationally efficient solutions. The findings indicate that using Bernoulli polynomials alongside improved block-pulse functions shows potential for solving complex integral equations with great accuracy.

This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of Hermite polynomials, Caputo fractional derivatives were computed efficiently, avoiding the complexities of direct fractional differentiation. The resulting weighted least-squares formulation transforms the problem into a stable algebraic system. Numerical results confirm the method’s superior accuracy, rapid convergence, and robustness compared with existing techniques, demonstrating its potential for broader applications in fractional-order boundary value problems.

This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two m-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions.

Many branches of practical mathematics rely heavily on numerical solutions to initial value problems, boundary value problems, and eigenvalue problems for ordinary and partial differential equations. Recently, the authors attempted to solve integral equations using hybrid Bernstein functions and improved block pulse functions. However, this is the first study to present a technical coupling between hybrid Bernstein and improved block-pulse functions for solving differential equations. The current method transforms differential equations into an algebraic system that can be solved with conventional methods. To validate the new method, certain numerical examples are supplied. The findings demonstrated that the method is both promising and highly accurate. The numerical findings reveal that the suggested hybrid approach outperforms the use of Bernstein polynomials, multi derivative hybrid block methods, block-pulse function, and other methods indicated in the numerical section in terms of accuracy. The proposed method can be implemented for more kinds of differential equations. The proposed method is applicable to a broader range of differential equations. The numerical results show that the proposed method are highly accurate and effective.

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 $\left(\mathcal{E}\mathcal{B}\mathcal{P}Fs\right)$. This suggested representation makes use of both the local support nature and computation efficiency of the $\left(\mathcal{E}\mathcal{B}\mathcal{P}Fs\right)$ 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 $\mathfrak{B}\mathcal{E}\mathcal{B}\mathcal{P}\mathcal{F}$ approach provides highly accurate approximations while requiring relatively small basis dimensions. Comparisons with previously reported numerical techniques further illustrate their computational effectiveness and accuracy.

In order to solve linear systems of Fredholm integral equations, this paper proposes a novel hybrid numerical method that combines improved block-pulse functions with Legendre polynomials. By utilizing the orthogonality and strong approximation properties of Legendre polynomials along with the computational simplicity of improved block-pulse functions, the suggested method converts the integral system into an equivalent system of algebraic equations. The approach outperforms a number of conventional numerical methods in terms of accuracy and convergence speed. The robustness, efficiency, and stability of the suggested scheme are validated by several numerical remarks, which also show how well it works for solving complex Fredholm integral systems that arise in scientific and engineering applications.