## Abstract

In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.

## Abstract

In this paper we study a nonlinear multi-dimensional partial differential equation, namely, a generalized second extended (3+1)-dimensional Jimbo-Miwa equation. We perform symmetry reductions of this equation until it reduces to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is obtained in terms of the Weierstrass zeta function. Also travelling wave solutions are derived using the simplest equation method. Finally, the conservation laws of the underlying equation are computed by employing the conservation theorem due to Ibragimov, which include conservation of energy and conservation of momentum laws.

## Abstract

In this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

## Abstract

Balaban index is defined as *G*, *n* and *m* are the cardinalities of the vertex and the edge set of *G*, respectively, and *w*(*u*) (resp. *w*(*v*)) denotes the sum of distances from *u* (resp. *v*) to all the other vertices of *G*. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each *n*, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.

## Abstract

Let *G* be a graph and let *m _{ij}*(

*G*)

*, i, j ≥*1, represents the number of edge of

*G*, where

*i*and

*j*are the degrees of vertices

*u*and

*v*respectively. In this article, we will compute different polynomials of flower graph

*f*(

*), namely*

_{n×m}*M polynomial*and

*Forgotten*polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.

## Abstract

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

## Abstract

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The *Brillouin* and *Langevin* as well as the *error* function, the *tanh*, *sigmoid* and the *tan ^{-}*

^{1}functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.

## Abstract

In this manuscript, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. We focus on four applications: optimal control applied to compartmental models, brain tumors, drug resistance and antiangiogenic treatment.

## Abstract

This paper presents an analytical method to determine the rise-set times of satellite-satellite visibility periods in different orbits. The Visibility function in terms of the orbital elements of the two satellites versus the time were derived explicitly up to *e*
^{4}. The line-of-sight corrected for Earth Oblateness up to *J*
_{2}, were considered as a perturbation to the orbital elements. The visibility intervals of the satellites were calculated for some numerical examples in order to test the results of the analytical work.

## Abstract

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions *K*
_{5} and *K*
_{12} by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.