Fully Legendre Spectral Galerkin Algorithm for Solving Linear One-Dimensional Telegraph Type Equation

2019 ◽  
Vol 16 (08) ◽  
pp. 1850118 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Type Equation ◽  
High Accuracy ◽  
Basis Functions ◽  
Structured Matrices ◽  
One Dimensional ◽  
Wide Applicability ◽  
Careful Investigation

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

scholarly journals Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 269-280 ◽  
Author(s):  
M.A. Abdelkawy ◽  
Engy A. Ahmed ◽  
Rubayyi T. Alqahtani
Keyword(s):  
Numerical Algorithm ◽  
Jacobi Polynomials ◽  
High Accuracy ◽  
Basis Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
One Dimensional ◽  
Numerical Tests ◽  
Shifted Jacobi Polynomials ◽  
Derivatives Of

AbstractWe introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs). We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

scholarly journals A Collocation Method for Numerical Solution of Hyperbolic Telegraph Equation with Neumann Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. C. Mittal ◽  
Rachna Bhatia
Keyword(s):  
Boundary Conditions ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Telegraph Equation ◽  
Neumann Boundary Conditions ◽  
Basis Functions ◽  
One Dimensional ◽  
First Order ◽  
Spline Basis ◽  
Good Agreement

We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.

scholarly journals ON SOME PROBLEMS FOR A LOADED PARABOLIC-HYPERBOLIC EQUATION

2017 ◽  
Vol 19 (6) ◽  
pp. 201-204
Author(s):  
A.V. Tarasenko
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Rectangular Area ◽  
Criterion Of Uniqueness

Some problems with various boundary conditions for the loaded mixed type equation in rectangular area are studied. The criterion of uniqueness is established and theorems of an existence of solutions to the problems are proved. The solutions are constructed as Fourier series with respect to eigenfunctions of a corresponding one-dimensional problem.

scholarly journals Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
M. M. Tharwat ◽  
A. H. Bhrawy ◽  
A. S. Alofi
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Sampling Theorem ◽  
High Accuracy ◽  
Internal Point ◽  
Derivative Sampling ◽  
Sturm Liouville ◽  
Truncation And Amplitude Errors ◽  
Hermite Approximation ◽  
Hermite Interpolations

The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.

ON THE INTERACTION OF A VISCOELASTIC CYLINDRICAL PIPE WITH INNER UNIFORM COATING AND INSERTS WITH A COMPLEX PROFILE

2020 ◽  
Author(s):  
Kirill Kazakov
Keyword(s):  
Complex Shape ◽  
High Accuracy ◽  
Contact Stresses ◽  
Basis Functions ◽  
Complex Profile ◽  
Cylindrical Pipe ◽  
Interference Fit ◽  
Uniform Coating ◽  
Special Basis ◽  
Insert Region

This work is devoted to the formulation and construction of an analytical solution to the problem of contact between a cylindrical viscoelastic aging pipe with an internal thin coating and an insert having a complex shape placed inside the pipe with an interference fit. In practice, the presence of such coatings is required, for example, to protect the main structure from aggressive external or internal environments, for its electrical insulation, etc. The manufacturing process of the inner coating determines its possible heterogeneity (dependence of properties on coordinates). An insert placed inside a pipe can have a complex profile that has a rapidly changing function. Taking these features into account is important when analyzing the stress-strain state of pipes with an internal coating. Using an approach based on the use of special basis functions and the type of solution, a representation for the contact stresses in the pipe in the region of the rigid insert is obtained. This approach makes it possible to distinguish functions that describe the properties of the inner coating and the shape of the outer profile of the insert in the form of separate terms and factors in the expression for the contact stresses in the insert region. Therefore, in order to achieve high accuracy when carrying out calculations, it is sufficient to restrict ourselves to a relatively small number of terms

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

Wavelets in the Solution of Nongray Radiative Heat Transfer Equation

1998 ◽  
Vol 120 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Y. Bayazitoglu ◽  
B. Y. Wang
Keyword(s):  
Transfer Equation ◽  
Solution Method ◽  
Radiative Heat ◽  
Radiative Transfer Equation ◽  
Basis Functions ◽  
Wavelet Basis ◽  
System Of Equations ◽  
Heat Transfer Equation ◽  
One Dimensional ◽  
The One

The wavelet basis functions are introduced into the radiative transfer equation in the frequency domain. The intensity of radiation is expanded in terms of Daubechies’ wrapped-around wavelet functions. It is shown that the wavelet basis approach to modeling nongrayness can be incorporated into any solution method for the equation of transfer. In this paper the resulting system of equations is solved for the one-dimensional radiative equilibrium problem using the P-N approximation.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

Sign in / Sign up

Export Citation Format

Share Document