SPSMAT: GNU Octave software package for spectral and pseudospectral methods

After delving through different books, we know that they concentrated on ODEs based on linear problems. The trouble is most ODE problems including almost all nonlinear ones can’t be solved analytically. A numerical one, however, can be obtained by computer. The very important advantage of numerical approximations is that they can be obtained for almost any ODE. In addition to the speed of these methods, the nature of numbers helps scrutinize results obtained numerically even apply different analysis method to a problem and compare them with one another. On another hand, it is not easy to compare two exact formulas, but it is beneficial to compare two numbers, even as numbers or visually in plots therifeten . That simply can be detected as a reason for designing a numerical package rather than the analytical one.

We never assume that we are familiar with everything that a user might want to do. A great feature of the SPSMAT toolbox is that it does not have a Graphical User Interface (GUI). Instead, the user is interacting directly with the functions on the command line or in scripts and is able to mix and build his new insight in order to solve a problem. Providing a GUI prevent the user to mix, match, and test the SPSMAT high-level functions experimentally. Data and code, in fact, are in the hands of the end users to reshape them suitable to their need.

MATLAB is widely known and used in the science communities including scientific computing. Although MATLAB has a rich feature set and flexibility, it is relatively and ridiculously expensive. MATLAB environment is a commercial and ”closed” product, thus MATLAB kernel and libraries cannot be modified nor freely distributed. To allow exchanging insights and contributing scientific research, both the toolbox and the platform on which a toolbox runs must be completely free.

Mathworks like other proprietary firms has its own policies that may not help you freely use and develop some packages.

That was maybe the ultimate motive to use an open source platform. In addition to it, as what an Iranian proverb saying one clap is not audible, we required the users extend the analyses with their own code and be exchanged between users, and between students and their supervisors, facilitating collaboration and knowledge over Github where they can even transfer their proper contribution with us.
Nonetheless, these are not such limitation for MATLAB users at all. They can even download the package and use it.

Due to the accuracy of fractional differential and integral equations in modeling various natural phenomena, fractional calculus has become the focus of many researchers. Fractional differential equations provide an outstanding instrument to describe the complex phenomena in fields of viscoelasticity, electromagnetic waves, diffusion equations, and so on. Moreover, the fractional order models of real systems are more sufficient in comparison with the integer order cases. Therefore, the field of fractional calculus has motivated the interest of researchers in various fields like physics, chemistry, engineering and even finance where all of their scientists and students need to work with numerical methods of solving these fractional equations. We broadly aim at this spectrum and wanted to produce a package for all of these practitioners.

The classical Jacobi polynomials, have been used extensively in mathematical analysis and practical applications, and play a pivotal role in the analysis and implementation of spectral and pseudospectral methods bhrawy2 . It is easily proven that the Jacobi polynomials are the polynomials arising as eigenfunctions of a singular Sturm–Liouville problem bhrawy2 . The usual spectral method by Legendre or Chebyshev approximation, and available only for non-singular problems on rectangular bounded domains. However, the general Jacobi method can be used in a wide array of problems bhrawy3 .
In addition to this, in this version, also for the simplicity’s sake, we cover Jacobi polynomials which are wide-ranging polynomials encompassing Gegenbauer (ultraspherical), Chebyshev, and Legendre polynomials. But, for those who are not into these polynomials, we also offered the derived subcategories including Gegenbauer, Chebyshev (all four kinds), and Legendre polynomials.

Over the recent years, many researchers have been interested in studying the properties of various equations and providing robust and accurate analytical and numerical methods for solving these equations and recorded their work in their published papers: what we have seen at this moment is to see and gather their works in one single work. Moreover, some of the high-qualified state-of-the-art methods and landmark works are done and analyzed perfectly that let us code them in the best shape that is readable and understandable, and specifically, beneficial. In addition to this, other works done on spectral methods are out-dated; like the work of Funaro/Matlab that is done in 1993 via Fortran (See Funaro package). What these packages lack are that at that time fractional calculus, operational matrices and Lagrangian bases had not been developed like today. The other FORTRAN package, PseudoPack 2000 is implemented by Don and Costas and is available in [PseudoPack package]. In this work parallel computers capability involved to solve problems and suffer the same misfortunes we issued here. In the newest work by Trefethen in Chebfun package, a huge and broad work done to refer a tool for researchers. But the light version, and more importantly, the specification for solving differential and integral equation in terms of spectral and pseudospectral was not its first goal.

SPSMAT, lo and behold, was created to address some of these issues.

As the operations in MATLAB and Octave are based on the concept of matrix, and in some extends their performance is optimal for matrices, we have given a matirx-based method in order to reach the best action of MATLAB or Octave. Another reason is the parallel computation that is said to be feasible and meaningful when it comes to matrix operations like multiplications and so forth. In this version, we have not considered matrix parallel computation methods, but in due course, we will work on it and report its relevant results.

The well-defined Spectral methods, by using computer power, have been developed rapidly through the recent years for the numerical solutions of ordinary/partial differential/integral equations. They have been providing a satisfying accuracy compared with other numerical methods and have wide applications in many mathematical problems. As we have realized, the spectral methods have been developed in different ways to provide accurate solutions for linear and nonlinear ordinary/partial differential/integral equations. Additionally, spectral methods have received considerable attention in dealing with various problems. As their most significant characteristics, they reduce problems to those of solving a system of algebraic equations; this makes them easier for solving. Also, they have excellent error properties and they offer exponential rates of convergence for smooth problems. Such methods have matured over the last 50 years and, in many cases, meet the robustness and computational efficiency standards required in practice.

In general, there are two ways to construct a polynomial approximation to the solution $y(x)$: One is to use an interpolating polynomial between the values $y(x_{j})$ at some points $x_{j}$. The other is to use a series expansion in terms of Lagrange or orthogonal polynomials like Jacobi polynomials. Here, the latter idea is pursued, and both the Jacobi and Lagrange Jacobi polynomials are constructed and the difference between employing them are also represented. In fact, the pseudospectral methods (here using Lagrange polynomials) are based on the spectral techniques. In pseudospectral methods, there are particularly two steps to achieve a numerical approximation for a differential equation. Firstly, a proper finite or discrete representation of the solution should be selected– this can be done by polynomial interpolation of the solution based on some appropriate nodes. On another hand, it is known that the Lagrange interpolation polynomial based on equidistant points does not yield a satisfactory approximation to general smooth functions. As a matter of fact, when the number of collocation points increases, interpolant polynomials notably diverge. However, satisfying results are obtained by relating the collocation points to the structure of classical orthogonal polynomials, such as the well-established Jacobi-Gauss-Lobatto points. That can be figured as a cogent reason for choosing these points in defining Lagrange polynomials. The use of global polynomials together with Gaussian quadrature collocation points (here Jacobi-Gauss-Lobatto points) is known to provide accurate approximations that converge exponentially for problems whose solutions are smooth kidderlatifi ; fokkerlatifi ; delkhosh ; delkhosh2 ; div .

This software package is intended for those who have a taste for solving ODEs, PDEs, integral equations and Optimal Controls. Practitioners with a not-so-deep understanding of Matlab or Octave can use this package without any difficulty. But basically this package is meant to be useful to solve different differential and integral equations; so it stipulates that the users have a grasp of Numerical analysis beforehand.

Undergraduate students taking a course in scientific computing and any relevant course can find this package as a test machine to check different things with orthogonal polynomials and develop their work with this package: this is your lightweight companion. In nutshell, whoever you are, we aim to increase your appreciation of this fundamental subject.