The Exact Solution to an Ablation Problem With Arbitrary Initial and Boundary Conditions

1984 ◽  
Vol 51 (4) ◽  
pp. 821-826 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Exact Solutions ◽  
Melting Temperature ◽  
Physical Interpretation ◽  
Existence And Uniqueness ◽  
Frictional Heating ◽  
Convective Motions ◽  
Initial And Boundary Conditions

The problem of ablation by frictional heating in a semi-infinite solid with arbitrarily prescribed initial and boundary conditions is investigated. The study includes all convective motions caused by the density differences of various phases of the materials. It is found that there are two cases: (i) ablation appears immediately and (ii) there is a waiting period of redistribution prior to ablation. The exact solutions of velocities and temperatures of both cases are derived. The solutions of the interfacial positions are also established. Existence and uniqueness of the solutions are examined and proved. The conditions for the occurrence of these two cases are expressed by an inequality. Physical interpretation of the inequality is explored. Its implication coincides with one’s expectation. Ablation appears only when the surface temperature is at or above the melting temperature.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

scholarly journals On the differential system govering flows in magnetic field with data inLp

1998 ◽  
Vol 21 (2) ◽  
pp. 299-305 ◽  
Author(s):  
Fengxin Chen ◽  
Ping Wang ◽  
Chaoshun Qu
Keyword(s):  
Magnetic Field ◽  
Initial Data ◽  
Boundary Conditions ◽  
Differential System ◽  
Existence And Uniqueness ◽  
Mhd Equations ◽  
The Earth ◽  
Initial And Boundary Conditions ◽  
Time Existence ◽  
The Magnetic Field

In this paper we study the system governing flows in the magnetic field within the earth. The system is similar to the magnetohydrodynamic (MHD) equations. For initial data in spaceLp, we obtained the local in time existence and uniqueness ofweak solutions of the system subject to appropriate initial and boundary conditions.

Ablation by Frictional Heating

1983 ◽  
Vol 50 (4a) ◽  
pp. 712-716 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Physical Interpretation ◽  
Frictional Heating ◽  
Constant Speed ◽  
Numerical Example ◽  
Similarity Transformations ◽  
Reduced Problem ◽  
As If

The ablation problem of a semi-infinite solid, moving at a constant speed parallel to its surface, is investigated. The study includes all induced motions caused by the density differences of various phases of the materials. Some appropriate transformations are introduced to reduce the problem to one where all phases behave as if they had the same density. The reduced problem is then solved by similarity transformations. It is found that the exact solution exists if and only if an inequality is satisfied. The physical interpretation of the inequality is examined. A numerical example is given to illustrate the result.

scholarly journals Improving WRF-Chem meteorological analyses and forecasts over aerosol polluted regions by incorporating NAAPS aerosol analyses

2021 ◽  
Author(s):  
Brittany N. Carson-Marquis ◽  
Jianglong Zhang ◽  
Peng Xian ◽  
Jeffrey S. Reid ◽  
Jared Marquis
Keyword(s):  
Boundary Conditions ◽  
Surface Temperature ◽  
Great Plains ◽  
Transport Model ◽  
Weather Prediction ◽  
The United States ◽  
Northern Great Plains ◽  
Chemical Transport Model ◽  
Near Surface ◽  
Initial And Boundary Conditions

AbstractWhen unaccounted for in numerical weather prediction (NWP) models, heavy aerosol events can cause significant unrealized biases in forecasted meteorological parameters such as surface temperature. To improve near-surface forecasting accuracies during heavy aerosol loadings, we demonstrate the feasibility of incorporating aerosol fields from a global chemical transport model as initial and boundary conditions into a higher resolution NWP model with aerosol-meteorological coupling. This concept is tested for a major biomass burning smoke event over the Northern Great Plains region of the United States that occurred during summer of 2015. Aerosol analyses from the global Navy Aerosol Analysis and Prediction System (NAAPS) are used as initial and boundary conditions for Weather Research and Forecasting with Chemistry (WRF-Chem) simulations. Through incorporating more realistic aerosol direct effects into the WRF-Chem simulations, errors in WRF-Chem simulated surface downward shortwave radiative fluxes and near-surface temperature are reduced compared with surface-based observations. This study confirms the ability to decrease biases induced by the aerosol direct effect for regional NWP forecasts during high-impact aerosol episodes through the incorporation of analyses and forecasts from a global aerosol transport model.

scholarly journals Exact solutions of time fractional heat-like and wave-like equations with variable coefficients

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 689-693 ◽  
Author(s):  
Sheng Zhang ◽  
Ran Zhu ◽  
Luyao Zhang
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Separation Method ◽  
Variable Separation ◽  
Science And Engineering ◽  
Special Solutions ◽  
Initial And Boundary Conditions ◽  
Variable Separation Method

In this paper, a variable-coefficient time fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag-Leffler function. As a result, exact solutions are obtained, from which some known special solutions are recovered. It is shown that the variable separation method can also be used to solve some others time fractional heat-like and wave-like equation in science and engineering.

scholarly journals An exact solution method for Fredholm integro-differential equations

2019 ◽  
pp. 2-8 ◽  
Author(s):  
Kyriaki Tsilika
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Solution Method ◽  
Initial Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Abstract Operator ◽  
Nonlocal Integral Boundary Conditions

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

scholarly journals A tale of two nested elastic rings

2017 ◽  
Vol 473 (2204) ◽  
pp. 20170340 ◽  
Author(s):  
G. Napoli ◽  
A. Goriely
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Variational Formulation ◽  
Contact Point ◽  
Elliptic Functions ◽  
Elastic Rods ◽  
Elastic Bodies ◽  
Static Solutions

Elastic rods in contact provide a rich paradigm for understanding shape and deformation in interacting elastic bodies. Here, we consider the problem of determining the static solutions of two nested elastic rings in the plane. If the inner ring is longer than the outer ring, it will buckle creating a space between the two rings. This deformation can be further influenced by either adhesion between the rings or if pressure is applied externally or internally. We obtain an exact solution of this problem when both rings are assumed inextensible and unshearable. Through a variational formulation of the problem, we identify the boundary conditions at the contact point and use the Kirchhoff analogy to give exact solutions of the problems in terms of elliptic functions. The role of both adhesion and pressure is explored.

The Stefan Problem of a Polymorphous Material

1979 ◽  
Vol 46 (4) ◽  
pp. 789-794 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Similarity Solution ◽  
Existence And Uniqueness ◽  
Series Solutions ◽  
Special Cases ◽  
Interfacial Boundary ◽  
Infinite Region ◽  
Uniformly Convergent

The problem of freezing or melting of a polymorphous material in a semi-infinite region with arbitrarily prescribed initial and boundary conditions is studied. Exact solutions of the problem are established. The solutions of temperature of all phases are expressed in polynomials and functions in the error integral family and time t and the position of the interfacial boundaries in power series of t1/2. Existence and uniqueness of the series solutions are considered and proved. It is also shown that these series are absolutely and uniformly convergent. The paper concludes with some remarks on density changes at the interfacial boundary and various special cases, one of which is the similarity solution.

scholarly journals Analytical exact solution of telegraph equation using HPM

BIBECHANA ◽  
2016 ◽  
Vol 14 ◽  
pp. 30-36
Author(s):  
Jamshad Ahmad ◽  
Ghulam Mohiuddin
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Partial Differential Equations ◽  
Telegraph Equation ◽  
Homotopy Perturbation

In this paper, exact solutions of different variants of second order hyperbolic telegraph equation are investigated with Homotopy Perturbation Method (HPM). The results determined by the proposed method are quite satisfactory and shows that HPM technique is very effective and useful for solving the nonlinear partial differential equations (PDEs) with given initial or boundary conditions. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions.BIBECHANA 14 (2017) 30-36

Exact solutions of fractional heat-like and wave-like equations with variable coefficients

2014 ◽  
Vol 24 (2) ◽  
pp. 455-467 ◽  
Author(s):  
Bo Tang ◽  
Xuemin Wang ◽  
Leilei Wei ◽  
Xindong Zhang
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Fractional Order ◽  
Variable Coefficient ◽  
Variational Iteration Method ◽  
Variable Coefficients ◽  
Mathematical Tool ◽  
Content Type ◽  
Fractional Differential ◽  
Initial And Boundary Conditions

Purpose – This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions. Design/methodology/approach – The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method. Findings – The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations. Originality/value – The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

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