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Weak Form of Peridynamic Equilibrium Equations
This chapter presents the weak form of peridynamic (PD) equilibrium equations to impose displacement and traction boundary conditions without a...
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On some mean field games and master equations through the lens of conservation laws
In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of...
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On the Equilibrium of Elastic Bodies with a Weakly Curved Junction
AbstractThe paper addresses the analysis of a boundary value problem with an unknown contact area that describes equilibrium of two-dimensional...
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Balance Equations
Borrowing from the mechanics of systems of particles, the balance equations for mass, linear momentum, and angular momentum are established. Next,...
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On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability
AbstractConditions for the strong ellipticity of equilibrium equations are formulated within strain gradient elasticity under finite deformations. In...
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Lagrange multiplier and variational equations in mechanics
The equilibrium of a structure is characterized by either Euler’s equations completed with boundary and some internal conditions, or by variational...
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Systems of Differential Equations
A system of ordinary differential equations consists of several equations containing derivatives of unknown functions of one variable.
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Factorization of Equilibrium Equation in Toupin–Mindlin Strain-Gradient Elasticity in Quaternion Analysis
In this paper, we introduce a factorization of the equilibrium equation in the theory of Toupin–Mindlin strain-gradient elasticity in the framework...
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Refined Equations of the Theory of Composite Multilayered Shells with Transversally Soft Core under Medium Bending
AbstractIn development of the results obtained earlier, an improved two-dimensional mathematical model of static and dynamic deformation of composite...
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Block-pulse integrodifference equations
We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing...
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A Survey of Results of St. Petersburg State University Research School on Nonlinear Partial Differential Equations. I
AbstractThe article contains a review of the most important results obtained in the framework of the St. Petersburg State University Research School...
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Solvability of Nonlinear Equilibrium Problems for Timoshenko-type Shallow Shells in Curvilinear Coordinates
AbstractWe prove the existence of solutions to the boundary value problem of equilibrium of elastic shallow inhomogeneous isotropic shells with free...
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The Asymptotic Behavior of the Reynolds Equations’ Solution
The work deals with the dynamic analysis of a mechanical system consisting of two rigid surfaces, one of which is in motion relative to the other. We...
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Partial Differential Equations
In the previous chapters of this book, we have seen that the ODEs provide a mathematical formulation for various problems in mechanics, physics, and...
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Introduction to Ordinary Differential Equations
This chapter is about the most basic concepts of the theory of differential equations. We will answer some fundamental questions: What is a...
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Fractional Differential Equations: A Primer for Structural Dynamics Applications
This paper presents a very brief overview of the framework for the application of fractional calculus (FC) in structural dynamics with the target...
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Mathematical Model and Numerical Simulation of Equilibrium Plasma Configurations in ‘‘Threeleaf’’ Magnetic Traps
AbstractIn this paper, the results of numerical studies of plasma configurations confined in equilibrium state by a magnetic field in the trap...
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Direct and Inverse Problems of Seismic Exploration of Anisotropic and Dispersive Elastic Media on Volume Integral Equations
AbstractThe theory of seismic exploration is based on the theory of elasticity, where an important role is played by material equations: Hooke’s law....
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On the Nonsymmetric Equilibrium Forms of Circular Plates under Normal Pressure
AbstractThe results of studying the bifurcation of axisymmetric equilibrium forms of circular plates under various conditions of fastening of the...
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Partial Differential Equations
We first build a second difference block matrix corresponding to the Laplacian on the square. We use this Laplacian matrix with various enforced...