scholarly journals A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mazhar Iqbal ◽  
M. T. Mustafa ◽  
Azad A. Siddiqui
Keyword(s):  
Gas Flow ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Similarity Solutions ◽  
Standard Application ◽  
Approximate Similarity ◽  
Initial Boundary Value ◽  
Unsteady Gas Flow

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

scholarly journals NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

2013 ◽  
Vol 18 (1) ◽  
pp. 80-96
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Rotating Magnetic Field ◽  
Droplet Dynamics ◽  
Ill Posed ◽  
Initial Boundary Value

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

scholarly journals Benjamin-Ono Equation on a Half-Line

2010 ◽  
Vol 2010 ◽  
pp. 1-38 ◽  
Author(s):  
Nakao Hayashi ◽  
Elena I. Kaikina
Keyword(s):  
Boundary Value Problem ◽  
Large Time ◽  
Boundary Value ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Half Line ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Boundary Value ◽  
Time Existence

We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Unsteady, Combined Radiation and Conduction in an Absorbing, Scattering, and Emitting Medium

1973 ◽  
Vol 95 (3) ◽  
pp. 357-364 ◽  
Author(s):  
K. C. Weston ◽  
J. L. Hauth
Keyword(s):  
Boundary Value Problem ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Gaussian Quadrature ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Isotropic Scattering ◽  
Radiation Parameter ◽  
Parallel Plates ◽  
Initial Boundary Value

The transient cooldown of a gray, absorbing, isotropic scattering, emitting, and conducting medium bounded by gray, diffusely emitting and reflecting parallel plates is considered. Numerical solutions are obtained for the initial boundary-value problem with a discontinuous decrease in temperature at one boundary. The quasi-steady equation of radiative transfer is solved using Gaussian quadrature and a matrix eigenvector technique together with explicit numerical solution of the unsteady energy equation. Temperature and energy flux distributions are presented for variations of optical thickness, boundary emissivity, albedo, and conduction–radiation parameter.

scholarly journals MATHEMATICAL MODELLING OF RF PLASMA AT LOW PRESSURES FOR CYLINDRIC VACUUM CHAMBER WITH CHARGED SPECIMAN

2022 ◽  
pp. 44-49
Author(s):  
Александр Юрьевич Шемахин ◽  
Виктор Семенович Желтухин ◽  
Евгений Юрьевич Шемахин
Keyword(s):  
Gas Flow ◽  
Vacuum Chamber ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rf Plasma ◽  
Neutral Gas ◽  
Potential Part ◽  
Potential Component ◽  
Initial Boundary Value

Для моделирования процессов в ВЧ-плазме пониженного давления с продувом газа разработана гибридная математическая модель при числах Кнудсена - для несущего газа. Модель включает начально-краевую задачу для кинетического уравнения Больцмана, описывающего функцию распределения несущего нейтрального газа, краевые задачи для уравнения неразрывности электронной, ионной и метастабильной компонент, уравнения сохранения энергии электронов, для ВЧ-уравнений Максвелла в форме телеграфных уравнений и уравнения Пуассона для потенциальной составляющей поля. Приводятся результаты расчета электрической напряженности, концентрации электронов, ионов и метастабилей, потенциальной составляющей электромагнитного поля в цилиндрической вакуумной камере. A hybrid mathematical model for the Knudsen numbers - for the carrier gas has been developed to simulate processes in a low pressure RF plasma with gas flow. The model includes an initial boundary value problem for the kinetic Boltzmann equation describing the distribution function of the carrier neutral gas, boundary value problems for the continuity equation of the electronic, ionic and metastable components, the electron energy conservation equations, for Maxwell’s RF equations in the form of telegraphic equations and the Poisson equation for the potential part of field. The results of the calculation of the electric intensity, the concentration of electrons, iones and metastables, the potential component of the electromagnetic field in a cylindrical vacuum chamber are presented.

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

Stochastic diffusion equation with fractional Laplacian on the first quadrant

2019 ◽  
Vol 22 (3) ◽  
pp. 795-806
Author(s):  
Jorge Sanchez-Ortiz ◽  
Francisco J. Ariza-Hernandez ◽  
Martin P. Arciga-Alejandre ◽  
Eduard A. Garcia-Murcia
Keyword(s):  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stochastic Evolution ◽  
Stochastic Diffusion Equation ◽  
Representation Of Solutions ◽  
Fokas Method ◽  
Initial Boundary Value ◽  
Main Ideas

Abstract In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

Evolution of assymetric swirl upward flows adjacent to vacuum

2020 ◽  
pp. 4-10
Author(s):  
S. P. Bautin ◽  
◽  
S. L. Deryabin ◽  
Keyword(s):  
Gas Dynamics ◽  
Gas Flow ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Equation System ◽  
Initial Time ◽  
Existence And Uniqueness Theorem ◽  
Contact Characteristic ◽  
Initial Boundary Value

The problems of modeling three-dimensional flows adjacent to vacuum were regarded earlier (see, for example, [1–5]). In works [1–3] onedimensional and multi-dimensional flows of polytropic and normal gas adjacent to vacuum were investigated. In works [4, 5] symmetric swirl upward flows of polytropic gas around of the vertical located contact characteristic separating the gas and vacuum were considered. It is shown that even in case the gas abuts vacuum, it swirls counterclockwise. It is also found that the vortex itself moves westward, shifting slightly northward. The present paper considers the evolution of the asymmetric gas flow at the initial time continuously adjacent to vacuum. An equation system of gas dynamics under the action of gravity and Coriolis force is adopted as a mathematical model. For the equation system of gas dynamics the initial boundary value problem is set on the multiplicity characteristic of four. The solution to the problem is created in the form of a power series and the existence and uniqueness theorem of the solution around the free surface «gas-vacuum» is proved.

On the Computations of Gas-Solid Mixture Two-Phase Flow

2014 ◽  
Vol 6 (01) ◽  
pp. 49-74 ◽  
Author(s):  
D. Zeidan ◽  
R. Touma
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Test Cases ◽  
Two Phase ◽  
Solid Mixture ◽  
Flow Problems ◽  
Model Equations ◽  
Initial Boundary Value

AbstractThis paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

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