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.

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 CONTINUATION FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS DISCRETIZED BY THE MULTIQUADRIC METHOD

2000 ◽  
Vol 10 (02) ◽  
pp. 481-492 ◽  
Author(s):  
A. I. FEDOSEYEV ◽  
M. J. FRIEDMAN ◽  
E. J. KANSA
Keyword(s):  
Basis Function ◽  
Elliptic Partial Differential Equations ◽  
High Accuracy ◽  
Basis Functions ◽  
Elliptic Pdes ◽  
Algebraic Equations ◽  
Nonlinear Elliptic ◽  
Interpolation Problems ◽  
Nonlinear Algebraic Equations ◽  
Meshless Collocation

The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.

scholarly journals One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

2021 ◽  
Author(s):  
M. Semplice ◽  
E. Travaglia ◽  
G. Puppo
Keyword(s):  
Boundary Conditions ◽  
The Novel ◽  
Two Dimensional ◽  
Finite Volume Schemes ◽  
Third Order ◽  
One Dimensional ◽  
Numerical Tests ◽  
Ghost Cells ◽  
The One ◽  
Appl Math

AbstractWe address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. 10.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

scholarly journals Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

Approximate methods of solving amplitude-phase problem for continuous signals

2021 ◽  
pp. 37-46
Author(s):  
Ilia V. Boikov ◽  
Yana V. Zelina
Keyword(s):  
Integral Equations ◽  
Collocation Methods ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Phase Problem ◽  
Spline Collocation ◽  
Approximate Methods ◽  
Nonlinear Operator Equations ◽  
One Dimensional

Amplitude and phase problems in physical research are considered. The construction of methods and algorithms for solving phase and amplitude problems is analyzed without involving additional information about the signal and its spectrum. Mathematical models of the amplitude and phase problems in the case of one-dimensional and two-dimensional continuous signals are proposed and approximate methods for their solution are constructed. The models are based on the use of nonlinear singular and bisingular integral equations. The amplitude and phase problems are modeled by corresponding nonlinear singular and bisingular integral equations defined on the numerical axis (in the one-dimensional case) and on the plane (in the two-dimensional case). To solve the constructed nonlinear singular and bisingular integral equations, spline-collocation methods and the method of mechanical quadratures are used. Systems of nonlinear algebraic equations that arise during the application of these methods are solved by the continuous method of solving nonlinear operator equations. A model example shows the effectiveness of the proposed method for solving the phase problem in the two-dimensional case.

scholarly journals Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Third Order ◽  
Generalized Jacobi Polynomials ◽  
Nonlinear Algebraic Equations ◽  
Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

scholarly journals Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
A. Atangana
Keyword(s):  
Jacobi Polynomials ◽  
Operational Matrix ◽  
High Accuracy ◽  
Computational Method ◽  
Singular Kernel ◽  
Variable Order ◽  
Fractional Differentiation ◽  
Established Method ◽  
Shifted Jacobi Polynomials ◽  
Fowler Equation

AbstractIn this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations

2017 ◽  
Vol 18 (5) ◽  
pp. 411-425 ◽  
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
Ali H. Bhrawy ◽  
José A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Volterra Integral Equations ◽  
The Novel ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Novel Technique ◽  
Shifted Jacobi Polynomials

AbstractThis paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE.

scholarly journals ISOGEOMETRIC COLLOCATION METHODS

2010 ◽  
Vol 20 (11) ◽  
pp. 2075-2107 ◽  
Author(s):  
F. AURICCHIO ◽  
L. BEIRÃO DA VEIGA ◽  
T. J. R. HUGHES ◽  
A. REALI ◽  
G. SANGALLI
Keyword(s):  
Theoretical Analysis ◽  
Isogeometric Analysis ◽  
Computational Cost ◽  
Basis Functions ◽  
Three Dimensions ◽  
Collocation Methods ◽  
One Dimensional ◽  
Numerical Tests ◽  
Low Computational Cost ◽  
Theoretical Results

We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

The pricing of average options with jump diffusion processes in the uncertain volatility model

2017 ◽  
Vol 04 (01) ◽  
pp. 1750005
Author(s):  
Yulian Fan ◽  
Huadong Zhang
Keyword(s):  
Diffusion Processes ◽  
Jump Diffusion ◽  
Two Dimensional ◽  
Discrete Solution ◽  
One Dimensional ◽  
Numerical Tests ◽  
Jump Diffusion Processes ◽  
Convergence Results ◽  
Volatility Model ◽  
Uncertain Volatility

The pricing equations of the average options with jump diffusion processes can be formulated as two-dimensional partial integro-differential equations (PIDEs). In the uncertain volatility model, for options with non-convex and non-concave payoffs, such as the butterfly spread, the PIDEs are nonlinear. We use the semi-Lagrangian method to reduce the two-dimensional nonlinear PIDE to a one-dimensional nonlinear PIDE along the trajectory of the average price, and use a Newton-type iteration to guarantee the convergence of the discrete solution to the viscosity solution. Monotonicity and stability as well as the convergence results are derived. Numerical tests of convergence for a variety of cases, including average butterfly spread and ordinary butterfly spread, are presented.

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