scholarly journals An observation problem for the Bessel differential operator

1984 ◽  
Vol 26 (1) ◽  
pp. 92-107 ◽  
Author(s):  
K.-D. Werner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Parabolic Partial Differential Equation ◽  
Final State ◽  
Bessel Differential Operator ◽  
State Observability

AbstractIn this paper, the parabolic partial differential equation ut = urr + (1/r)ur − (v2/r2)u, where v ≥ 0 is a parameter, with Dirichlet, Neumann, and mixed boundary conditions is considered. The final state observability for such problems is investigated.

scholarly journals Boundary value control problems involving the bessel differential operator

1986 ◽  
Vol 27 (4) ◽  
pp. 453-472 ◽  
Author(s):  
K.-D. Werner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Necessary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Finite Energy ◽  
Control Problems ◽  
Hyperbolic Partial Differential Equation ◽  
Boundary Controllability ◽  
Bessel Differential Operator

AbstractIn this paper, we consider the hyperbolic partial differential equation wrr = wrr + 1/r wr − ν2 /r2w, where v ≥ 1/2 or ν = 0 is aprameter, with the Dirichlet, Neumann and mixed boundary conditions. The boundary controllability for such problems is investigated. The main resutl is that all “finite energy” intial states can be steered to the zero state in time T, using a control f ∈ L2 (0, T), provided T > 2. Furthermore, necessary conditions for controllability are also presented.

scholarly journals INTERVAL EXTENSION STRUCTURE BASIC TYPES OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

2018 ◽  
pp. 10-14
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Interval Function ◽  
The Third ◽  
Interval Extension ◽  
The Neumann Problem

. In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the Rfunction method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where 𝑃 = {𝜔𝛷 , 𝜔𝛷̅, 𝜔̅𝛷, 𝜔̅𝛷̅} и [ 𝛷, 𝛷̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form. For the problem of the third type, the solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [𝛷2, 𝛷2̅̅̅̅] -indefinite, interval function, 𝐷1 - differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure,[ 𝛷1, 𝛷1], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form.

On nonlinear mixed boundary value problems for second order elliptic differential equations on domains with corners

1980 ◽  
Vol 87 (1-2) ◽  
pp. 35-51 ◽  
Author(s):  
Bernhard Kawohl
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Regularity Of Solutions ◽  
Singular Boundary ◽  
Elliptic Differential Equations ◽  
Linear Differential ◽  
Domains With Corners

SynopsisWe investigate the existence, uniqueness and regularity of solutions to the linear differential equation Lu = f under nonlinear mixed boundary conditions on domains with singular boundary points.

scholarly journals A Data-Driven Based Spatiotemporal Model Reduction for Microwave Heating Process with the Mixed Boundary Conditions

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 827
Author(s):  
Jiaqi Zhong ◽  
Shan Liang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Model Reduction ◽  
Microwave Heating ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Data Driven ◽  
Heating Process ◽  
Spatiotemporal Model ◽  
Time Space

In this paper, a data-driven based spatiotemporal model reduction approach is proposed for predicting the temperature distribution and developing the computation speeds in the microwave heating process. Due to the mixed boundary conditions, it is difficult for the traditional spectral method to directly obtain the analytical eigenfunctions. Motivated by the time/space separation theory, we first propose a general framework of spatiotemporal model reduction, which can effectively develop the computation speeds in the numerical analysis of multi-physical fields. Subsequently, the empirical eigenfunctions are generated by applying the Karhunen–Loève theory to decompose the snapshots. Then, the partial differential Equation (PDE) model is discretized into a class of recursive equations and transformed as the reduced-order ordinary differential Equation (ODE) model. Finally, the effectiveness and superiority of the proposed approach is demonstrated by a comparison study with a traditional method on the microwave heating Debye medium.

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