The well-known equations for the powder compaction process (PCP) in a rigid die published from the beginning of the last century until today were considered in this review. Most of the considered equations are converted into the dependences of densif...
Unsaturated titanium hydride (TiHX) powder has high formability and is a promising raw material for titanium-based powder metallurgy. In this work, TiH2, TiHX, and HDH Ti powders were characterized, the cold compaction behavior of the powders was inv...
Equations of plasticity of a porous body proposed by different authors and obtained under the condition that the yield surface of a porous body has the shape of an ellipsoid of revolution are considered in this paper. Such equations have two independ...
Based on the generalization of M. Yu. Balshin’s well-known equations in the framework of a discrete model of powder compaction process (PCP), two new die-compaction equations for powders have been derived that show the dependence of the compact...
Numerical solution and parameter estimation for a type of fractional diffusion equation are considered. Firstly, the symmetrical compact difference scheme is applied to solve the forward problem of the fractional diffusion equation. The stability and...
Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in th...
This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-or...
A method is proposed for the derivation of new classes of staggered compact derivative and interpolation operators. The algorithm has its roots in an implicit interpolation theory consistent with compact schemes and reduces to the computation of the...
This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time d...
In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in...
In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral t...
In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is der...
In this paper, high-order compact-difference schemes involving a large number of mesh points in the computational stencils are used to numerically solve partial differential equations containing high-order derivatives. The test cases include a linear...
The time-fractional Cattaneo equation is an equation where the fractional order α∈(1,2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an eq...
In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2&m...
In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinea...
In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The inte...
In this paper, high-order compact difference methods (HOCDMs) are proposed to solve the semi-linear Sobolev equations (SLSEs), which arise in various physical models, such as porous media flow and heat conduction. First, a two-level numerical method...
High-precision finite difference (FD) wavefield simulation is one of the key steps for the successful implementation of full-waveform inversion and reverse time migration. Most explicit FD schemes for solving seismic wave equations are not compact, w...
A high-order compact (HOC) implicit difference scheme is proposed for solving three-dimensional (3D) unsteady reaction diffusion equations. To discretize the spatial second-order derivatives, the fourth-order compact difference operators are used, an...
This work integrates the fast Alikhanov method with a compact scheme to solve the time-fractional Kuramoto–Sivashinsky (KS) equation with the generalized Burgers’ type nonlinearity. Initially, the Alikhanov algorithm, designed to handle t...
In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserv...
In this article, we present an efficient numerical strategy for the two-dimensional nonlinear Schrödinger equation, focusing on its development and analysis. Our approach begins with proposing a nonlinear, energy-conservative, fourth-order, comp...
Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic inter...
We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in three nonlinear models for surface gravity waves: the nonlinear Schrödinger equation, the super compact Zakharov equ...
The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the t...
The densification mechanism of Cu–Al mixed metal powder during a double-action die compaction was investigated by numerical simulation. The finite element method and experiment were performed to compare the effect of the forming method, such as...
A finite element model based on elastic–plastic theory was conducted to study the densification process of iron-based powder metallurgy during high velocity compaction (HVC). The densification process of HVC at different heights was simulated u...
This paper introduces a new measure of non-compactness within a bounded domain of RN in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in gener...
This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried...
We parameterize the core of compact spherical star configurations by a mass (mx) and a radius (rx) and study the resulting admissible areas in the total-mass–total-radius plane. The employed fiducial equation-of-state models of the corona at ra...
In the present work, a highly efficient incompressible flow solver with a semi-implicit time advancement on a fully staggered grid using a high-order compact difference scheme is developed firstly in the framework of approximate factorization. The fo...
In this paper, we present new existence theorems of mild solutions to Cauchy problem for some fractional differential equations with delay. Our main tools to obtain our results are the theory of analytic semigroups and compact semigroups, the Kuratow...
This paper presents an innovative numerical technique for specific classes of stochastic heat equations. Our approach uniquely combines a sixth-order compact finite difference algorithm with fast discrete Fourier transforms. While traditional discret...
(1) Background: This paper is devoted to the study of a class of semilinear initial boundary value problems of parabolic type. (2) Methods: We make use of fractional powers of analytic semigroups and the interpolation theory of compact linear operato...
A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is...
In the present paper, our main work aims to discover the existence result of the fractional order non-linear Hadamard functional integral equations on [1,a] by employing the theory of measure of non-compactness together with the fixed point theory in...
This article introduces new sufficient conditions ensuring the interior approximate controllability of semilinear thermoelastic plate equations subject to Dirichlet boundary conditions. The analysis is carried out by reformulating the system as an ab...
The existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions are investigated in this study. The fractional integro-differen...
This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (κ,ϕ)-Riemann–Liouville along with Erdélyi–Kober fractional...
In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvabil...
In this paper, we extend Darbo’s fixed point theorem via weak JS-contractions in a Banach space. Our results generalize and extend several well-known comparable results in the literature. The technique of measure of non-compactness is the main...
In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus Theory,...
The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function F:X→cc...
In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo deriv...
In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of nume...
This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by...
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and secon...
The primary objective of this study is to investigate the concept of approximate controllability in fractional evolution equations that involve the ψ-Caputo derivative. Specifically, we examine the scenario where the semigroup is compact and anal...
Compacted bentonite is envisaged as engineering buffer/backfill material in geological disposal for high-level radioactive waste. In particular, Na-bentonite is characterised by lower hydraulic conductivity and higher swelling competence and cation e...
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