Rajiniganth Pandurangan

Integro-differential equations (IDEs) seamlessly blend differential and integral operators, offering a versatile mathematical framework for modelling various physical, biological, and engineering systems. Analytical solutions to these equations are often unattainable due to their complexity, necessitating efficient numerical approaches. This study introduces the Legendre Collocation Method (LCM), leveraging shifted Legendre polynomials to approximate solutions for Volterra and Fredholm IDEs. By transforming the IDEs into a system of linear equations, the proposed method achieves high accuracy and computational efficiency. Numerical experiments on representative IDEs validate the approach, showing excellent agreement with exact solutions and outperforming existing methods in terms of absolute error. These results underscore the potential of the LCM in solving complex IDEs with broad applicability across disciplines.

In this study, we present a generalized higher-order delta operator with the co-efficient of falling factorial and its inverse, both of which allow us to get more generalized k(ε)-Fibonacci sequences along with their sums, a few theorems, and some intriguing conclusions regarding the sum of more generalized terms of k(ε)-Fibonacci sequences with falling factorial. In addition, the n-fold Fibonacci ratio and a few applications in life sciences and crystal growth were presented in the work. In addition, suitable examples are presented using MATLAB to show our results.

The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort’s formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete proportional Laplace transform and determine the GDPLT of various functions using the inverse operator. The results obtained are showcased through relevant examples and validated using MATLAB.

In this study, the Volterra-FredholmIntegro-Differential Equations (VFIDEs) are solved computationally using first-kind Chebyshev polynomials as basis functions. The technique converts VFIDEs into a system of unknown Chebyshev coefficients linear algebraic equations, which are then solved via matrix inversion. Five numerical examples were solved in order to assess the scheme's correctness and effectiveness. The findings reveal that the approach outperforms the comparison results in the literature

In this paper, we introduce second order difference operator with polynomial factorial and its inverse by which we obtain advanced Fibonacci sequence and its sum. Some theorems and interesting results on the sum of the terms of second order Fibonacci sequence are derived. Suitable examples are provided to illustrate our results and verified by MATLAB.