This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.

‎In this paper‎, ‎we delve into a coefficient inverse problem linked to the bioheat equation‎, ‎a pivotal component in medical research concerning phenomena such as temperature response and blood perfusion during surface heating‎. ‎By considering factors like heat transfer between tissue and blood in capillaries and incorporating the geometric intricacies of the skin‎, ‎we confine our analysis to a one-dimensional domain‎. ‎Our approach involves transforming the original problem into one concerning the reconstruction of a multiplicative source term within a parabolic equation‎. ‎Subsequently‎, ‎we utilize integral conditions to derive a specific integro-differential equation‎, ‎accompanied by the requisite initial and boundary conditions‎. ‎Leveraging a spectral method‎, ‎we streamline the modified problem into a linear system of algebraic equations‎. ‎To accomplish this‎, ‎we employ appropriate regularization algorithms to obtain stable approximations for the derivatives of perturbed boundary data and to effectively solve the resultant system of equations‎.

‎This paper presents a numerical approach for reconstructing the leading coefficient in an inverse heat conduction problem (IHCP)‎. ‎We consider a one-dimensional heat equation with known input data‎, ‎including the initial condition‎, ‎a supplementary temperature measurement at the final time‎, ‎and two integral observations‎. ‎By incorporating the terminal condition‎, ‎the unknown spatially dependent coefficient is eliminated‎, ‎reducing the problem to a nonclassical parabolic equation‎. ‎The unknown temperature distribution and its derivatives are approximated and applied to the modified governing equation‎, ‎which is then discretized using operational matrices of differentiation‎. ‎To ensure stable derivative estimation‎, ‎the method is coupled with a regularization technique‎. ‎A least squares scheme is employed to formulate a nonlinear system of algebraic equations‎, ‎which is solved using Newton’s method‎. ‎The reliability of the proposed solution is demonstrated through several numerical examples‎.

Although Elzaki transform is stronger than Sumudu and Laplace transforms to solve the ordinary differential equations withnon-constant coefficients, but this method does not lead to finding the answer of some differential equations. In this paper, a method is introduced to find that a differential equation by Elzaki transform can be ‎solved?‎

‎In this paper, the Lie symmetry method and Lie brackets of vector fields are used in order to find some new solutions of the (3+1)-dimensional sourceless wave equation‎. ‎The obtained solutions are classified into two categories; polynomial and non-polynomial exact solutions‎. ‎Because of the properties of the Lie brackets and the symmetries‎, ‎a generalized method is implemented for constructing new solutions from old solutions‎. ‎We demonstrate the generation of such polynomial and non-polynomial solutions through the medium of the group theoretical properties of the equation‎. ‎It is noteworthy that this method could be used when the equations have two special kinds of symmetries which will be mentioned below‎.

In this study‎, ‎we explore soliton solutions for the conformable time-fractional Boussinesq equation utilizing the three-wave method‎. ‎To validate the precision of our findings‎, ‎we discuss specific special cases by adjusting certain potential parameters and also present the graphical representations of our results‎. ‎The results achieved in this research align closely with those from previous studies‎, ‎demonstrating enhanced accuracy and simplicity‎. ‎Given the extensive applications of this equation in particle physics‎, ‎understanding its dynamics is crucial‎. ‎Consequently‎, ‎employing methods that encompass a broad spectrum of solutions is imperative‎. ‎The versatility of this method in yielding diverse solutions is evident in the results we have obtained‎. ‎The solutions derived in this paper are novel and offer greater precision compared to previous works‎.

‎    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.

‎The pantograph equation improves the mathematical model of the system includes modeling the motion of the wire connected with the dynamics of the supports and modeling the dynamics of the pantograph‎. ‎The subject of this paper is the existence and Ulam stability of solutions for a coupled system of sequential pantograph equations of fractional order involving both Riemann-Liouville and Caputo-Hadamard fractional derivative operators‎. ‎By applying the classical theorems in nonlinear analysis‎, ‎such as the Banach's fixed point theorem and Leray-Schauder nonlinear alternative‎, ‎the uniqueness and existence of solutions are obtained‎. ‎Furthermore‎, ‎the Ulam stability results are also presented‎. ‎Finally‎, ‎we have shown the results in the applications section by presenting various examples to numerical effects which provided to support the theoretical findings.

‎In this paper‎, ‎we consider a Sturm-Liouville equation with non-separated boundary conditions on a finite interval‎. ‎We discuss some properties of solutions of the Sturm-Liouville equation‎, ‎where the potential function has a singularity in the finite interval‎. ‎We also calculate eigenvalues and prove the uniqueness of Borg's Theorem of this boundary value problem‎.

In the field of advertising‎, ‎there are always situations in which individuals or companies promote their products in order to find retrieval opportunities and attract customers in a competitive environment‎. ‎Several goals are followed in this paper‎. ‎First‎, ‎the historical development of applications of differential games in modeling strategic situations in competitive advertising is mentioned‎. ‎We then introduce the problem in a duopoly market under the influence of uncertainty in the framework of a stochastic differential game‎. ‎Finding the equilibrium strategy for this problem requires solving a nonlinear partial differential equations system also known as the Hamilton-Jacoby equation‎. ‎Another purpose of this paper is to propose an efficient and appropriate computational method for solving the Hamilton-Jacobi partial differential equations‎. ‎The proposed method for solving the problem is a combination of collocation methods by the derivative operator matrix based on Chelyshkov polynomials and policy iteration method‎. ‎The advantage of using the policy iteration method is that at each step‎, ‎instead of finding the solution to a nonlinear partial differential equation‎, ‎it is sufficient to solve a sequence of linear partial differential equations systems‎. ‎The convergence of the proposed method is provided in detail‎. ‎Finally‎, ‎we solve the corresponding Hamilton-Jacobi equations system by the proposed iterative algorithm‎.