In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator...
Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. M...
In this paper, we consider a partial differential equation with mixed derivatives of first order in time and second order in the spatial variable. Such equations are usually referred to as one-dimensional pseudoparabolic equations. We prove the exist...
The purpose of this paper is to establish solvability results for both direct and inverse problems associated with the diffusion equation involving the Dunkl–Laplacian operator in a punctured domain. We demonstrate the existence and uniqueness...
The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems ar...
In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear...
In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. Th...
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. M...
In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain...
In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral...
In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental mo...
In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency...
The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, w...
This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from...
In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and
This study addresses the theoretical development and computational implementation of probabilistic-statistical methods for solving the stationary problems of two-phase filtration of two immiscible, incompressible fluids in a porous medium. The object...
Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicompon...
Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these eq...
In this paper, we study the question of constructing explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables in all coordinate directions. Such equations are a natural generalization of classica...
In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that th...
We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse different...
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential...
The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construc...
Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When
All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time
In this paper, using the technique of operator matrix, we consider the positive solution of the system of operator equations
This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by space–time symmetry breakage. Classi...
This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in
The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the...
In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation,
Fourier Neural Operators have become a central tool for learning solution operators of partial differential equations, but their spectral layers remain entirely classical and rely on digital Fourier processing. In this work, we introduce the Continuo...
In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fract...
In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden cond...
The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo...
A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equation...
Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for s...
This article presents a criterion for the uniqueness of the solution of a problem nonlocal in time for a differential-operator equation with a symmetric operator part on space variables. The symmetry of the operator part of the operator-differential...
The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differe...
In this paper, we used Fourier Neural Operator (FNO) networks to solve reaction–diffusion equations. The FNO is a novel framework designed to solve partial differential equations by learning mappings between infinite-dimensional functional spac...
This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is k...
In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring...
Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their pr...
In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the ph...
Consider a class of coupled discrete-time Riccati equations arising from jump systems. To compute their solutions when systems reach a steady state, we propose an operator Newton method and correspondingly establish its quadratic convergence under su...
In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and t...
This study investigates the existence of mild solutions to impulsive fractional differential equations involving almost-sectorial operators. Through the application of solution operator techniques, fixed-point theory, and Laplace transform, we demons...
In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the f...
In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves...
An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence o...
This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions:...
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