Mandal, Moumita

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods.

This article focuses on finding the approximate solutions of fractional Volterra integro-differential equations with non-smooth solutions using the shifted Jacobi spectral Galerkin method (SJSGM) and its iterated version. To deal with the singularity present in the kernel of the transformed weakly singular Volterra integral equation, we convert it into an equivalent weakly singular Fredholm integral equation. We first directly apply our proposed methods to this equivalent transformed equation and obtain improved convergence results by incorporating the singularity of the kernel function into the shifted Jacobi weight function. Further, we introduce a smoothing transformation and discuss the regularity of the transformed solution, and achieve superconvergence results for all γ∈(0,1). Additionally, we obtain super-convergence results for classical first-order Volterra integro-differential equations. Finally, numerical examples with a comparative study are provided to validate our theoretical results and verify the efficiency of the proposed methods. We show that the convergence rates can be obtained to the desired degree by increasing the value of the smoothing index ϱ (1<ϱ∈N), where N stands for the set of natural numbers. © 2024 Elsevier B.V.

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results. © 2024 IMACS

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non-monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order (Formula presented.), including non-monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function (Formula presented.), with (Formula presented.) being a positive real number, the value (Formula presented.) is a turning point when (Formula presented.). However, for (Formula presented.), the turning point shifts to the left of point (Formula presented.) and shifts to the right of point (Formula presented.) for (Formula presented.) We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis.

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results. © 2024 Elsevier Inc.