scholarly journals Trefftz Method of Solving a 1D Coupled Thermoelasticity Problem for One- and Two-Layered Media

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3637
Author(s):  
Artur Maciąg ◽  
Krzysztof Grysa
Keyword(s):  
Coupled System ◽  
Approximate Solutions ◽  
Thermoelasticity Problem ◽  
Successive Approximations ◽  
Trefftz Method ◽  
Coupled Thermoelasticity ◽  
General Solutions ◽  
Basic Equations ◽  
The Right ◽  
Derivatives Of

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the problem under consideration. The solutions for one layer and for a two-layer plate are graphically presented and analyzed, illustrating the possible application of the method. Our results show that increasing the number of Trefftz functions leads to the reduction of differences between successive approximations.

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

Approximate Solutions in Linear, Coupled Thermoelasticity

1968 ◽  
Vol 35 (2) ◽  
pp. 255-266 ◽  
Author(s):  
R. E. Nickell ◽  
J. L. Sackman
Keyword(s):  
Half Space ◽  
Rapid Heating ◽  
Ritz Method ◽  
Gaussian Quadrature ◽  
Boundary Surface ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Coupled Thermoelasticity ◽  
Initial Boundary Value ◽  
Thermally Conducting

A method for obtaining approximate solutions to initial-boundary-value problems in the linear theory of coupled thermoelasticity is developed. This procedure is a direct variational method representing an extension of the Ritz method. As an illustration of the procedure, it is applied to a class of one-dimensional, transient problems involving weak thermal shocks. The problems considered are: (a) Rapid heating of a half space through a thermally conducting boundary layer, and (b) gradual heating of the boundary surface of a half space. The solutions generated by the extended Ritz method are compared, for accuracy, to solutions obtained from a numerical inversion scheme for the Laplace transform based on Gaussian quadrature. These comparisons indicate that the variational procedure developed here can yield accurate results.

scholarly journals Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zaid Laadjal ◽  
Qasem M. Al-Mdallal ◽  
Fahd Jarad
Keyword(s):  
Boundary Conditions ◽  
Coupled System ◽  
Fractional Integrals ◽  
Langevin Equations ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Nonseparated Boundary Conditions ◽  
New Type ◽  
Predictor Corrector ◽  
Derivatives Of

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

scholarly journals Determine of the wave equation in the task of electrical oscillations

2018 ◽  
Vol 184 ◽  
pp. 01023
Author(s):  
Gordana V. Jelić ◽  
Vladica Stanojević ◽  
Dragana Radosavljević
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Mathematical Physics ◽  
Equation Of Motion ◽  
Independent Variables ◽  
Equations Of Mathematical Physics ◽  
Basic Equations ◽  
Derivatives Of ◽  
Two Independent Variables ◽  
Transverse Oscillations

One of the basic equations of mathematical physics (for instance function of two independent variables) is the differential equation with partial derivatives of the second order (3). This equation is called the wave equation, and is provided when considering the process of transverse oscillations of wire, longitudinal oscillations of rod, electrical oscillations in a conductor, torsional vibration at waves, etc… The paper shows how to form the equation (3) which is the equation of motion of each point of wire with abscissa x in time t during its oscillation. It is also shown how to determine the equation (3) in the task of electrical oscillations in a conductor. Then equation (3) is determined, and this solution satisfies the boundary and initial conditions.

A fast meshfree technique for the coupled thermoelasticity problem

2018 ◽  
Vol 229 (6) ◽  
pp. 2657-2673 ◽  
Author(s):  
Kourosh Hasanpour ◽  
Davoud Mirzaei
Keyword(s):  
Thermoelasticity Problem ◽  
Coupled Thermoelasticity

Approximations by multivariate perturbed neural network operators

2017 ◽  
Vol 15 (03) ◽  
pp. 413-432 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Neural Network ◽  
Rate Of Convergence ◽  
Activation Function ◽  
Jackson Type ◽  
Partial Derivatives ◽  
Right Hand ◽  
Hidden Layer ◽  
The Right ◽  
Derivatives Of

This article deals with the determination of the rate of convergence to the unit of each of three newly introduced here multivariate perturbed normalized neural network operators of one hidden layer. These are given through the multivariate modulus of continuity of the involved multivariate function or its high-order partial derivatives and that appears in the right-hand side of the associated multivariate Jackson type inequalities. The multivariate activation function is very general, especially it can derive from any multivariate sigmoid or multivariate bell-shaped function. The right-hand sides of our convergence inequalities do not depend on the activation function. The sample functionals are of multivariate Stancu, Kantorovich and quadrature types. We give applications for the first partial derivatives of the involved function.

A preclinical survey on the efficacy of NSC-290205 in combination with doxorubicin (ADR) in Lewis lung carcinoma (LLC)

2006 ◽  
Vol 24 (18_suppl) ◽  
pp. 17131-17131
Author(s):  
A. Papageorgiou ◽  
E. Stergiou ◽  
I. Boukovinas ◽  
G. Geromichalos ◽  
I. Stergiou
Keyword(s):  
Antitumor Activity ◽  
Tumor Growth ◽  
Maximum Effect ◽  
Base Pairs ◽  
Chop Regimen ◽  
C57bl Mice ◽  
Inhibition Of Tumor Growth ◽  
Reduced Toxicity ◽  
The Right ◽  
Derivatives Of

17131 Background: NSC-290205 (A) is a hybrid synthetic antitumor ester, which combines a D-lactam derivative of androsterone and nitrogen mustard. Studies on modified steroidal esters of carboxylic derivatives of N,N-bis(2-chloroethyl)aniline, have shown that they exhibit reduced toxicity and increased antitumor activity and specificity. In this study we investigated the antitumor activity of compound A in combination with ADR (AHOP) in comparison with standard CHOP regimen. Methods: C57Bl mice were used for the antitumor evaluation of AHOP/CHOP. Experiments were initiated by implanting the tumor. LLC cells (purchased by NCI, Bethesda, USA) were implanted intramuscularly into the right hind leg as a suspension of 7 × 106 cells in 0.1 ml. The antitumor activity was assessed from the inhibition of tumor growth by volume in cm3 and the oncostatic parameter T/C % according to the protocol of experimental evaluation of antitumor drugs of the NCI, USA. Treatments were given as a single dose (D) on day 1, intermitted dose (D/2 × 3) on days 1, 5, 9 or consecutive dose (D/4 × 9) on days 1 through 9. Results: Treatment with A or cyclophosphamide produced almost equal borderline activity. Moreover, both CHOP and AHOP regimens showed significant and comparable antitumor effect (p < 0.05 by the Wilkoxon test). AHOP caused the maximum effect inhibiting the tumor growth by 67.7% and T/C values of 270%. CHOP was less effective producing 54.8% tumor inhibition and T/C values of 238%. Conclusions: It is very likely that the D-lactamic steroid (androstan) alkylator for A, containing the -NHCO group, combined with ADR, which intercalates between base-pairs, is the explanation for higher activity of AHOP vs. CHOP. This significant effect of NSC-290205 with the anthracycline adriamycin on LLC adds to NSC-290205 advantage for further clinical development. No significant financial relationships to disclose.

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