scholarly journals The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

2015 ◽  
Vol 3 ◽  
pp. 186
Author(s):  
Ērika Teirumnieka ◽  
Ilmārs Kangro ◽  
Edmunds Teirumnieks ◽  
Harijs Kalis
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Elliptic Type ◽  
Exponential Spline ◽  
Initial Boundary Value

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

Travelling waves in an initial-boundary value problem

1981 ◽  
Vol 90 (1-2) ◽  
pp. 41-61 ◽  
Author(s):  
E. J. M. Veling
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Expression ◽  
Travelling Waves ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Travelling Wave ◽  
Boundary Values ◽  
Initial Boundary Value

SynopsisIn this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials

1989 ◽  
Vol 113 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
Nicolas Charalambakis ◽  
François Murat
Keyword(s):  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Dependent ◽  
Initial Boundary Value

SynopsisWe prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

scholarly journals On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

2019 ◽  
Vol 21 (4) ◽  
pp. 413-429
Author(s):  
Pavel N. Burago ◽  
Albert I. Egamov
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Phase Constraint ◽  
Initial Boundary Value

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

scholarly journals Integro-differential systems with variable exponents of nonlinearity

2017 ◽  
Vol 15 (1) ◽  
pp. 859-883 ◽  
Author(s):  
Oleh Buhrii ◽  
Nataliya Buhrii
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Existence Theorem ◽  
Fourth Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Exponents ◽  
Differential Systems ◽  
Initial Boundary Value

Abstract Some nonlinear integro-differential equations of fourth order with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.

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