###### Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis

## Abstract

With the development of modern partial differential equation (PDE) theory, the theory of linear PDE is becoming more and more perfect, . Non-linear PDE has become a research hotspot of many mathematicians. In fact, when describing practical physical problems with PDEs, non-linear problems tend to be more general than linear problems, which are close to real problems and have practical physical significance. Hyperbolic PDEs are a kind of important PDEs describing the phenomena of vibration or wave motion. The solution of hyperbolic PDE can be decomposed into the form of multiplication of vibration and vibration or of exponential function and exponential function. Generally, the energy is infinite. A full discrete convergence analysis method for non-linear hyperbolic equation based on finite element analysis is proposed. Taking second-order and fourth-order non-linear hyperbolic equation as examples, the full discrete convergence of non-linear hyperbolic equation is analysed by finite element method and the super-convergence results are obtained.

###### Investigation of A Fuzzy Problem by the Fuzzy Laplace Transform

## Abstract

This paper is on the solutions of a fuzzy problem with triangular fuzzy number initial values by fuzzy Laplace transform. In this paper, the properties of fuzzy Laplace transform, generalized differentiability and fuzzy arithmetic are used. The example is solved in relation to the studied problem. Conclusions are given.

###### A New Approach For Weighted Hardy’s Operator In VELS

## Abstract

A considerable number of research has been carried out on the generalized Lebesgue spaces *L*
^{p(x)} and boundedness of different integral operators therein. In this study, a new approach for weighted increasing near the origin and decreasing near infinity exponent function that provides a boundedness of the Hardy’s operator in variable exponent space is given.

###### Numerical Solution of Abel′s Integral Equations using Hermite Wavelet

## Abstract

A numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods.

###### Soluble Product of Parafree Lie Algebras and Its Residual Properties

## Abstract

The parafree Lie algebras are an extraordinary class of Lie algebras which shares many properties with a free Lie algebra. In this work, we turn our attention to soluble product of parafree Lie algebras. We show that soluble product of parafree Lie algebras is parafree. Furthermore, we investigate some residual properties of that product.

###### An Allee Threshold Model for a Glioblastoma(GB)-Immune System(IS) Interaction with Fuzzy Initial Values

## Abstract

In this paper, we adopt the model of [12] by including fuzzy initial values to study the interaction of a monoclonal brain tumor and the macrophages for a condition of extinction of GB(Glioblastoma) by using Allee threshold. Numerical simulations will give detailed information on the behavior of the model at the end of the paper. We perform all the computations in this study with the help of the Maple software.

###### A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

## Abstract

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.

The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, 𝓛-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number *h*(*d*) formula in real quadratic fields claims that we have

for positive *d* > 0 and the fundamental unit *ε _{d}* of ℚ(

*ε*are significant and necessary tools for determining the structure of real quadratic fields.

_{d}The focus of this paper is to determine structure of some special real quadratic fields for *d* > 0 and *d* ≡ 2, 3 (*mod* 4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element *w _{d}*, fundamental unit

*ε*, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

_{d}###### Multidimensional BSDE with Poisson jumps of Osgood type

## Abstract

This paper is devoted to solve a multidimensional backward stochastic differential equation with jumps in finite time horizon. Under linear growth generator, we prove existence and uniqueness of solution.

###### QSPR Analysis of certain Distance Based Topological Indices

## Abstract

In QSAR/QSPR study, topological indices are utilized to guess the bioactivity of chemical compounds. In this paper, we study the QSPR analysis of selected distance and degree-distance based topological indices. Our study reveals some important results which help us to characterize the useful topological indices based on their predicting power.

###### New Exact Solutions for Generalized (3+1) Shallow Water-Like (SWL) Equation

## Abstract

In this study, we use the improved Bernoulli sub-equation function method for exact solutions to the generalized (3+1) shallow water-like (SWL) equation. Some new solutions are successfully constructed. We carried out all the computations and the graphics plot in this paper by Wolfram Mathematica.