105 Results Found

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new exist...

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techni...

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods,...

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving fam...

High order convective Cahn-Hilliard type equations describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we prove the well-posedness of the classical solutions for the...

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Non...

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space ${\mathbb{R}}^m$. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of...

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on...

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The nume...

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Ca...

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are construct...

This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We...

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu fo...

The fifth order Kudryashov–Sinelshchikov equation models the evolution of the nonlinear waves in a gas–liquid mixture, taking into account an interphase heat transfer, surface tension, and weak liquid compressibility simultaneously at the...

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal c...

The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant...

In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the spa...

In this study, we investigate a class of mixed fractional partial integro-differential equations (FrPI-DE) involving symmetric singular kernels. The considered model problem involves Caputo fractional derivatives and integral operators that describe...

Unlike the uniform eigenstrain case, polynomial eigenstrain distributions capture spatially varying transformation fields, offering higher physical fidelity and direct relevance to the equivalent inclusion method (EIM) for non-elliptical inhomogeneit...

This article studies the abstract discrete-time Cauchy problem involving the Riemann–Liouville type difference operator. Sufficient conditions for the existence of unique solution to the semilinear Cauchy problem in Lebesgue and weighted Lebesg...

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order $0<\alpha <2$. The problem also constitutes a class of examples of the Cauchy proble...

In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) a...

A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect...

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the $\psi$-Hilfer derivative. We redefine the transform in a weighted setting, establis...

The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order $\alpha \in (0,1)$ and a Bolza-type cost functional. A relationship between the differenti...

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of t...

In this article, a thin infinite flexible plate weakened by multiple curvilinear holes is considered. The strength shapes are mapped outside a unit circle with the assistance of particular conformal mapping under certain conditions. The mathematical...

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with b...

The aim of this paper is to generalize the classical Filippov lemma to the framework of Cauchy differential set-valued problems involving the supremum of the unknown function on a past interval and its Stieltjes derivative with respect to a left-cont...

We consider material bodies exhibiting a response function for free energy, which depends on both the strain and its gradient. Toupin–Mindlin’s gradient elasticity is characterized by Cauchy stress tensors, which are given by space-like Euler–Lagrang...

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the resul...

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in ${\mathbb{R}}^{\mathrm{n}}$, highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-o...

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular a...

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equ...

This paper is related to the blow-up results of solutions to the Cauchy problem of semilinear generalized Tricomi equations, which contain a scale-invariant damping term and a mass term. The nonlinear term is of the power type in the case of a single...

We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, t...

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize feature...

We investigate linear and nonlinear poroacoustic waveforms under the Rubin–Rosenau– Gottlieb (RRG) theory of generalized continua. Working in the context of the Cauchy problem, on both the real line and the case with periodic boundary con...

This paper is devoted to new propagation and regularity results for a class of first-order Hamilton–Jacobi–Bellman-type problems in a separable infinite-dimensional Hilbert space. Specifically, the related Cauchy problem is investigated b...

A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently b...

The aim of this paper is to study the stability of generalized Liouville–Caputo fractional differential equations in Hyers–Ulam sense. We show that three types of the generalized linear Liouville–Caputo fractional differential equat...

In the paper, a fundamental solution of the fractional dual-phase-lagging heat conduction problem is obtained. The considerations concern the 1D Cauchy problem in a whole-space domain. A solution of the initial-boundary problem is determined by using...

This paper is concerned with the existence of solutions of a class of Cauchy problems for hyperbolic partial fractional differential inclusions (HPFD) involving the Caputo fractional derivative with an impulse whose right hand side is convex and non-...

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in ${\mathbb{R}}^d(d\ge 3)$. And the global well-posedness and Gevrey analytic of strong solutio...

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous...

The integro-differential equation with the Cauchy kernel is used in many different technical problems, such as in circuit analysis or gas infrared radiation studies. Therefore, it is important to be able to solve this type of equation, even in an app...

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our cu...

The study of the mixed initial-boundary value problem, corresponding to the thermoelasticity of porous micromorphic materials under the influence of microtemperatures, represents the main objective of this article. Achieving qualitative results on th...

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type mu...

Given an injective closed linear operator A defined in a Banach space $X,$ and writing ${}_{CF}{D}_t^{\alpha }$ the Caputo–Fabrizio fractional derivative of order $\alpha \in (0,1),$ we show that the unique solution of the abstract Cauchy problem $(\ast ){}_{CF}$...