Moving Node Method for the Approximate Analytical Solution One-dimensional Convection-Diffusion Problems| Chapter 4 | Novel Research Aspects in Mathematical and Computer Science Vol. 7

 The following four groups of approaches can be used to solve mathematical physics problems.

Analytical procedures, such as the method of characteristics, the method of Green's functions, etc., have a relatively low degree of universality and are therefore concentrated on resolving relatively small classes of issues.

Analytical methods are less ubiquitous than approximate analytical methods, which include projection, variational methods, small parameter methods, operational methods, and different iterative methods.

Numerical techniques, such as the finite difference method, method of lines, control volume technique, and finite element technique, are extremely versatile techniques.

Monte Carlo methods, often known as probabilistic approaches, are quite flexible. Calculations for discontinuous solutions are possible. However, when solving such issues to which these approaches are suitable, they necessitate a lot of calculations and typically fall short of the computational complexity of the above methods. This chapter discusses novel methods for resolving boundary value issues in differential equations. It brings a fresh approach to node movement. Analytical solutions that are roughly accurate are produced by approximating differential equations (using the finite difference approach or the control volume method) and incorporating the idea of a moving node. Multipoint moving nodes are employed to improve the accuracy of the analytical answers that are obtained. Compact circuits are built using the moving node technique. You can examine the monotonicity of the diskette equation and the differential equation's approximation error using the moving node approach. Different test issues are taken into account. Mathematical subjects.

Author(s) Details:

Dalabaev Umurdin,
Department of Mathematics Modeling and Informatics, University of World Economy and Diplomacy, Tashkent, Uzbekistan.

Ikramova Malika,
Scientific Research Institute of Irrigation and Water Problems, Uzbekistan.

Umarova Shoira,
Department of Mathematics Modeling and Informatics, University of World Economy and Diplomacy, Tashkent,

Uzbekistan.

Please see the link here: https://stm.bookpi.org/NRAMCS-V7/article/view/7949

Some Stable Difference Schemes to Describe the Flow in the Combined Region | Chapter 10 | Theory and Practice of Mathematics and Computer Science Vol. 9

 Flows occur in a saturated porous fluid medium in many environmental, industrial, and biological processes. The transport of substances from surface water to groundwater is a major issue. The mathematical model is based on the interpenetrating model of two-phase media (Rahmatullin model). The proposed equations allow for a uniform analysis of the flow of a liquid in and out of the porous zone. The Navier-Stokes equation is obtained in the liquid region in this case. The equations in the porous region are very similar to the Brinkman model. There is no need to set boundary conditions in the separation area (such as Beavers – Joseph – Saffman) when describing the flow from the perspective of a single equation for the entire region. If the porous region's energy calculation is used, cross-border conditions occur. The order of the systems of equations in each field is different in this case. The energy estimate for the Rahmatullin equation is derived using energy inequalities based on the proposed model. The difference scheme of Rahmatullin's equation is constructed, and the constructed scheme's stability is determined.

Author (s) Details

U. Dalabaev
The University of World Economy and Diplomacy, Buyuk Ipak Yuli Str., Tashkent, Uzbekistan.

View Book :- https://stm.bookpi.org/TPMCS-V9/article/view/857

Critical Study on Finite Element Solution of a Boundary Value Problems for Equation Gravity Gyroscopic Waves in the Time Domain | Chapter 8 | Recent Studies in Mathematics and Computer Science Vol.4

 In the present work, on the base of a numerical finite element method, the solution of the Dirichlet and

Neumann problems with respect to the oscillation equation for gravity-gyroscopic waves is discussed. The
approximation with respect to spatial variables is achieved by using linear splines, and the approximation with
respect to time is achieved by using cubic Hermitean splines. It is demonstrated that the use of such
approximation with respect to time allows the quality of the solution to be essentially improved as compared
with the traditional approximation ensuring the second order accuracy. The stability and accuracy of the method
are estimated. Using the method of regularization with spectrum shift, a new method is developed for solving
the spatial operator degeneration problem associated with the Neumann problem. The results of the numerical
calculations performed provide the possibility to make conclusions on the mode of behavior of the solution of
the Neumann problem depending on the problem variables.

Author(s) Details

Mikhail Nikolayevich Moskalkov
Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

Dauletbay Utebaev
Berdakh Kara-Kalpak State University, Nukus, Uzbekistan.

View Book :- http://bp.bookpi.org/index.php/bpi/catalog/book/234