An application of the Exp-function method (EFM) to search for EXACT solutions of nonlinear partial differential equations is analyzed. This method is used for the MODIFIED KdV equation and the generalized KdV equation. The EFM was used to construct peri- odic wave and solitary wave solutions of nonlinear evolution equations (NLEEs). This method is developed for searching EXACT travelling wave solutions of nonlinear partial differential equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics and applied mathematics.

In this paper, by applying the non-classical symmetry method, nonclassical symmetries of the Kodryashov-Sinleschikov (K-S) and MODIFIED Korteweg-de Vries-Zaharov-Kuznetsov (mKdV-ZK) equations are obtained. Apart from classical symmetries, this theory can be effective in finding a few other solutions for a system of PDEs and ODEs. Also, non-classical symmetries of a system of PDEs can be applied to reduce the number of independent variables. By adding the invariance surface condition to the assumed equations and applying the classical symmetry method for them, non-classical symmetries are calculated. Finally, some of the group invariant solutions and the similarity reduced equations associated to non-classical symmetry are obtained.

‎    This paper introduces a MODIFIED version of the Variational Iteration Method, incorporating $\mathbb{P}$-transformation. We propose a novel semi-analytical technique named the MODIFIED variational iteration method   for addressing fractional differential equations featuring tempered Liouville-Caputo derivatives. The MODIFIED variational iteration method emerges as a highly efficient and powerful mathematical tool, offering EXACT or approximate solutions for a diverse range of real-world problems in engineering and the natural sciences, specifically those expressed through differential equations. To assess its effectiveness and accuracy, we scrutinize the MODIFIED variational iteration method by applying it to three problems related to the heat-like multidimensional diffusion equation with a fractional time derivative in a tempered Liouville-Caputo form.

An analytical investigation of the buckling problem of composite shells with radius variation is presented. Different types of loading such as compressive and external can be applied. Flügge’s shell equations, MODIFIED for anisotropic laminated materials are used. The modal forms are assumed to have axial dependency in the form of simple Fourier series. To implement the present method to find the buckling load of composite tubes with different boundary conditions the derivatives of Fourier series are legitimized using Stocke’s transformation. The mathematical model presented includes the radius variations at the cross section of tubes in the form of functions including the imperfection factor. The analytical procedure developed in this work for finding the buckling loads yields an EXACT equation which is simpler than the EXACT method adopted by other research workers. The results are presented in the form of buckling diagrams and figures showing the mode shapes.

In this paper, we consider the classical Lie symmetries of fractional MODIFIED Zakharov Kuznetsov (which in this paper we abbreviately show this by the mZK equation) equation. Indeed, Lie symmetries are utilized for solving the nonlinear fractional three-dimensional mZK equation with partial derivatives, and by using the infinitesimal transformations and corresponding invariant solutions, we reduce the underlying equation one dimension less than the original mZK equation, and finally, some of the corresponding EXACT solutions are extracted. Indeed, Lie groups are geometric powerful tools for analyzing and investigating a wide variety of classes of equations such as ordinary differential equations, partial differential equations, fractional differential equations, and integral and integro differential equations. Invariant solutions and conservation laws that play a very significant and astonishing role in physical science can be obtained by this method. Moreover, various kinds of this method such as classical, non-classical, approximate and et cetera can be extracted by utilized in this field.

This text carries out some ideas about EXACT and d P− EXACT sequences of transformation semi groups. Some theorems like the short five lemma (Lemma 1.3 and Lemma 2.3) are valid here as in EXACT sequences of R − modules.

Variance of EWMA Statistic is needed when control chart is used. In literatures, a relation is used for computation of this variance which is not EXACT. We show that this variance is less than the EXACT value. This paper first derives the EXACT variance of the EWMA statistic for independent observations. Then through the comparison of this new variance and old variance, the effect of EXACT variance on control limits of EWMA control chart is examined.

Let λ be an infinite regular cardinal and A a locally λ-presentable additive category. We show that any λ-pure morphism (resp. λ-pure quotient) in A creates a kernel-cokernel pair. This implies that the class of all λ-pure kernel-cokernel pairs in A forms an EXACT structure. Additionally, we will describe λ-pure kernel-cokernel pairs in A and will prove that any λ-directed diagram of objects in A induces a canonical λ-pure kernel-cokernel pair.

Identification of the EXACT location of an exon in a DNA sequence is an important research area of bioinformatics. The main issues of the previous signal processing techniques are accuracy and robustness for the EXACT locating of exons. To address the mentioned issues, in this study, a method has been proposed based on deep learning. The proposed method includes a new preprocessing, a new mapping method, and a multi-scale MODIFIED and hybrid deep neural network. The proposed preprocessing method enriches the network to accept and encode genes at any length in a new mapping method. The proposed multi-scale deep neural network uses a combination of an embedding layer, a MODIFIED CNN, and an LSTM network. In this study, HMR195, BG570, and F56F11.4 datasets have been used to compare this work with previous studies. The accuracies of the proposed method have been 0.982, 0.966, and 0.965 on HMR195, BG570, and F56F11.4 databases, respectively. The results reveal the superiority and effectiveness of the proposed hybrid multi-scale CNN-LSTM network.