In this paper, a family of numerical methods that uses collocation and interpolation techniques is developed to solve Volterra Integro-Differential Equations (VIDEs). The basic idea involved is the reduction of the VIDEs to Initial Value Problems (IVPs), and then solved using functionally-fitted Adam’s methods in a block-by-block mode. To achieve this, the proposed method is constructed through the techniques of interpolation and collocation. The numerical properties, such as zero-stability and region of stability of the proposed methods, are also investigated. Some numerical examples from recent scientific literature are used to demonstrate the efficiency and validity of the new methods.
In a recent article published in Boundary Value Problems the authors established some multiplicative inequalities of the Hermite–Hadamard–Mercer and trapezoidal-Mercer type involving the Riemann–Liouville fractional integral operators. In this paper under consideration, we generalize some of these results and add to the literature novel estimates containing the κ -Riemann–Liouville fractional integral operators. We introduce a new technique in establishing these results which we believe is an interesting contribution to this subject area. Additionally, we develop a new auxiliary lemma and use it to establish two new multiplicative trapezoidal-Mercer type inequalities. We also examine how the fractional parameter affects the improvement and refinement of the resulting estimates. To illustrate the theoretical results, we present several numerical examples together with computational and graphical analyses. The results obtained in this study contribute to the ongoing development of multiplicative fractional integral inequalities and may be helpful for further exploration in generalized convexity, multiplicative calculus, and related areas of fractional analysis.
We investigate a Viscous Holographic Dark Energy (VHDE) model in Brans–Dicke [1] theory using the Marder metric, which belongs to the Bianchi type-I class. The field equations are solved by imposing suitable relations among the metric potentials together with a prescribed Hubble parameter, which yields a constant deceleration parameter q=m-1 . Thus, the model describes accelerated or decelerated expansion depending on the value of m . The cosmological behavior of VHDE is examined through its energy density, equation-of-state parameter, stability via the squared speed of sound, and other cosmological diagnostics.
