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This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results.

The results of this paper are theoretical and analytical in nature.

The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results.

Analytical and theoretical results.

The results of this paper can be applied in science and engineering.

The results of this paper is applicable in certain social sciences.

The results of this paper are new and will open up new areas of research in mathematical sciences.

The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and △-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.

In this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.

Showed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.

This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

The purpose of this paper is to study the coupled fixed point problem and the coupled best proximity problem for single-valued and multi-valued contraction type operators defined on cyclic representations of the space. The approach is based on fixed point results for appropriate operators generated by the initial problems.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.

The results of this paper are theoretical and analytical in nature.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.

The results are theoretical and analytical.

The results were applied to solving nonlinear integral equations.

The results has several social applications.

The results of this paper are new.

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.