Homogeneous Equation Research Articles

Boundary value problems for discontinuously loaded parabolic equations

The article deals with boundary value problems for a discontinuously loaded parabolic equation with a Riemann – Liouville fractional integro-differentiation operator with variable coefficients. The unambiguous solvability of the Cauchy – Dirichlet problem for a discontinuously loaded parabolic equation of fractional order is proved. The paper also examines the existence and uniqueness of the solution of the first boundary value problem for a discontinuously loaded parabolic equation. Using the method of the Green function, using the properties of the fundamental solution of the corresponding homogeneous equation, as well as assuming that the coefficients of the equation are bounded, continuous and satisfy the Helder condition, while remaining non-negative, it is shown that the solution of the problem is reduced to a system of Volterra integral equations of the second kind.

Criterion of the Existence of a Strongly Continuous Resolving Family for a Fractional Differential Equation with the Hilfer Derivative

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 families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of C0-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed.

Increasing the Accuracy of Solving Boundary Value Problems with Linear Ordinary Differential Equations Using the Bubnov-Galerkin Method

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

On the theory of generalized analytic Vekua functions with a singular point

This work consists of two sections. The first section examines the elliptic system of equations ∂ z ¯ u + a ( z ) u + b ( z ) u ¯ = f . The concept of Vekua spaces (V-spaces) is introduced, namely: a Banach function space B is a Vekua space if the theorem about continuity of the solution of an inhomogeneous equation and the theorem about the representation of solutions of a homogeneous equation holds for any functions a ( z ) , b ( z ) and f ( z ) . The following main statement is obtained (Theorem 7.1): A function space B possessing the properties 1 ∘ − 3 ∘ of § 2 is a V – space if and only if B ↪ P 1 (for the definition of P 1 see § 2). From this result it follows that a symmetric space is a V space if and only if it is continuously embedded in the Lorentz space L ( 2 , 1 ) (Theorem 7.3). Thus, we can say that the widest space to which the Vekua theory can be extended is P 1 ( ⋅ ) , and among all symmetric spaces - L ( 2 , 1 ) . In the second section, we study the following equation with singular point ∂ z ¯ u + a ( z ) | z | u + b ( z ) | z | u ¯ = f . The paper proves the existence of solutions to this equation in the functional class B introduced in the first part of the article. Representations of the 1st and 2nd types of solutions from this class are obtained.

ADER-WAF Schemes for the Homogeneous One-Dimensional Shallow Water Equations

ADER-WAF Schemes for the Homogeneous One-Dimensional Shallow Water Equations

The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations

The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations

On the exponential type explosion for solution of 3D modified critical homogeneous convective Brinkman-Forchheimer equation

On the exponential type explosion for solution of 3D modified critical homogeneous convective Brinkman-Forchheimer equation

On p-self semi homogeneous system of difference equations of dimension two

This article introduces a novel notion concerning a system of homogeneous difference equations (^E): a P-Self-Semi-Homogeneous System (PSSHS) of order m, denoting a positive integer for both p and m. The article differentiates between two states (p=1 and p≠1, respectively) and two substrates (m=1 and m≥2), where m represents a value for each state. A matrix containing coefficients of the identical dimension (2×2) was used to summarize the work. The argument is substantiated by a compilation of definitions, findings, evidence that constitute the body of our article. In conclusion, instances and resolutions for every scenario are presented, each of which can be resolved through one of three distinct methods: application of the definition, application of the result, or utilization of the theorem.

On Some Cauchy Problems and Positive Linear Operators

Durrmeyer type modifications of the Szász–Mirakjan operators were studied in several papers. We consider a general family of such operators. Basically, we investigate the relationship between the images of a function f under the operators and the solutions of a suitable linear and homogeneous partial differential equation of parabolic type with f as initial value.

Advanced Dynamic Thermal Vibration of Thick Composited FGM Cylindrical Shells with Fully Homogeneous Equation by Using TSDT and Nonlinear Varied Shear Coefficient

A numerical method using advanced nonlinear shear is used to study the thermal vibration of functionally graded material (FGM) thick circular cylindrical shells. The third-order shear deformation theory (TSDT) of displacements is applied and the equations are derived of the motion of cylindrical shells and the expression of the advanced nonlinear varied shear factor. The expressions of stiffness of thick composited two-layer FGM circular cylindrical shells with sinusoidal rising temperature are applied. The partial differential equation (PDE) in dynamic equilibrium of thick FGM circular cylindrical shells is derived with respect to shear rotations and displacements under terms of thermal–mechanical loads and density inertia terms. Important parametric effects of the advanced nonlinear varied shear factor, power law index, and temperature on the stress and displacement of thick FGM circular cylindrical shells are studied. Additionally, the advanced nonlinear varied shear factor effect is included and studied for a vibrating frequency using a fully homogeneous equation.

Modeling of shear wave propagation in piezo-viscoelastic microbeam over quadratic heterogeneous viscoelastic plate with sliding contact

Abstract This study analyzes the phase and attenuation dynamic behavior of piezo-viscoelastic microbeam overlying quadratic heterogeneous viscoelastic plate under sliding contact. Using the Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Maxwell’s relations are used to incorporate the electric potential function. The solutions for both media are derived separately by solving the second-order hyperbolic differential equation using the method of separation of variables and asymptotic expansions of Bessel functions. The system of linear homogeneous equations is obtained by applying admissible boundary conditions to determine fundamental physical quantities. The key contribution of the current work is demonstrating the influence of dissipation factors, sliding contact, micro-length, heterogeneity, and thickness ratio parameters on shear wave propagation. The micro-length effect is found to suppress the attenuation of shear waves through the analysis and discussion of the dispersion and attenuation curves.

An asymptotic result for linear nonhomogeneous mixed type differential equation

We consider a nonhomogeneous linear mixed type differential equation with variable coefficients and establish an asymptotic result for the solutions. Our result is obtained by the use of a solution of the so-called generalized characteristic equation of the corresponding homogeneous linear mixed type differential equation.

Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies. II. Fully Analytical Equations and Pseudoadvective Eigenenergies

Physical insight into plasma evolution in the magnetohydrodynamic (MHD) limit can be revealed by decomposing the evolution according to the characteristic modes of the system. In this paper we explore aspects of the eigenenergy decomposition method (EEDM) introduced in an earlier study (ApJ, 967:80). The EEDM provides an exact decomposition of nonlinear MHD disturbances into their component eigenenergies associated with the slow, Alfvén, and fast eigenmodes, together with two zero-frequency eigenmodes. Here we refine the EEDM by presenting globally analytical expressions for the eigenenergies. We also explore the nature of the zero-frequency “pseudoadvective (PA) modes” in detail. We show that in evolutions with pure advection of magnetic and thermal energy (without propagating waves), a part of the energy is carried by the PA modes. Exact expressions for the error terms associated with these modes—commonly encountered in numerical simulations—are also introduced. The new EEDM equations provide a robust tool for the exact and unique decomposition of nonlinear disturbances governed by homogeneous quasi-linear partial differential equations, even in the presence of local or global degeneracies.

$\mu$-Stability of Positive Homogeneous Differential-Difference Equations With Unbounded Time-Varying Delays

$\mu$-Stability of Positive Homogeneous Differential-Difference Equations With Unbounded Time-Varying Delays

Dynamic analysis of the air–water interface instability in a wind-wave flume: Initial conditions for wave generation

This paper aims to investigate the dynamic mechanism for a previously calm water surface under a background wind field, which would lose stability due to minor perturbations. When spatial distribution and growth rate of the shear wind field satisfy a certain relationship, an Orr–Sommerfeld equation can be derived from linearized Navier–Stokes equations, for the amplitude of perturbation waves at the air–water interface. After transforming its coordinates to a finite interval, this fourth-order linear ordinary differential equation with variable coefficients can turn into a linear singular differential equation. It is solved under the background flow fields of air and water using the corrected Fourier method that we previously developed, to yield two sets of four analytical solutions with physical significance. The perturbation flow at the air–water interface must satisfy the kinematic and dynamic matching conditions (continuity of velocity and smooth transition of shear stress), which results in a fourth-order homogeneous linear algebraic equation for four undetermined constants. Setting various parameters in accordance with measured data with a wind-wave flume, the corresponding perturbation waves solution is identified. Our calculation densely distributes the most unstable and readily growing waves among perturbation waves with small phase speeds, consistent with the experimental results. Our perturbation solutions over a flat air–water interface are only applicable to the short period after instability onset, which provides an initial condition for wave generation. The consequent interaction between small-amplitude waves and wind field is beyond the scope of our linear theory, which has indeed been sufficiently addressed in the literature.

On the derivation of the homogeneous kinetic wave equation

On the derivation of the homogeneous kinetic wave equation

Dynamics of vortex cap solutions on the rotating unit sphere

Dynamics of vortex cap solutions on the rotating unit sphere

Long wavelength limit of the KP-type equation for the ion dynamics system

In this paper, we consider the modified Kadomtsev–Petviashvili equation (KP-type) limit for ion dynamics system in R 2 + 1 . We derive the two-dimensional homogeneous and inhomogeneous KP-type equation from the non-dimensional ion dynamics system by Gardner–Morikawa transforms. The different Gardner–Morikawa transforms in temporal–spatial directions bring the anisotropic problem for energy estimate. By the energy method, we prove that the solution of Euler–Poisson system converges to KP-type equation globally when ϵ → 0 in R 2 + 1 .

Some Exact Green Function Solutions for Non-Linear Classical Field Theories

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

The maxcut of the sunrise with different masses in the continuous Minkoskean dimensional regularisation

The maxcut of the sunrise with different masses in the continuous Minkoskean dimensional regularisation