scholarly journals Analogues of the Laplace Transform and Z-Transform with Piecewise Linear Kernels

Foundations ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 99-115
Author(s):  
Marianito R. Rodrigo ◽  
Mandy Li
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Piecewise Linear ◽  
The Laplace Transform

Two new transforms with piecewise linear kernels are introduced. These transforms are analogues of the classical Laplace transform and Z-transform. Properties of these transforms are investigated and applications to ordinary differential equations and integral equations are provided. This article is ideal for study as a foundational project in an undergraduate course in differential and/or integral equations.

scholarly journals The Laplace transform as an alternative general method for solving linear ordinary differential equations

2021 ◽  
Vol 1 (4) ◽  
pp. 309
Author(s):  
William Guo
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Case Studies ◽  
Initial Conditions ◽  
Linear Ordinary Differential Equations ◽  
The Laplace Transform ◽  
Linear Odes ◽  
General Method

<p style='text-indent:20px;'>The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.</p>

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Multi-term fractional integro-differential equations in power growth function spaces

2021 ◽  
Vol 24 (3) ◽  
pp. 739-754
Author(s):  
Vu Kim Tuan ◽  
Dinh Thanh Duc ◽  
Tran Dinh Phung
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Growth Function ◽  
Function Spaces ◽  
Power Growth ◽  
The Laplace Transform

Abstract In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.

HYPERCHAOS IN A MODIFIED CANONICAL CHUA'S CIRCUIT

2004 ◽  
Vol 14 (01) ◽  
pp. 221-243 ◽  
Author(s):  
K. THAMILMARAN ◽  
M. LAKSHMANAN ◽  
A. VENKATESAN
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Laboratory Experiments ◽  
System Parameter ◽  
Piecewise Linear ◽  
Electrical Circuit ◽  
First Order ◽  
Regular Behavior ◽  
Linear Elements

In this paper, we present the hyperchaos dynamics of a modified canonical Chua's electrical circuit. This circuit, which is capable of realizing the behavior of every member of the Chua's family, consists of just five linear elements (resistors, inductors and capacitors), a negative conductor and a piecewise linear resistor. The route followed is a transition from regular behavior to chaos and then to hyperchaos through border-collision bifurcation, as the system parameter is varied. The hyperchaos dynamics, characterized by two positive Lyapunov exponents, is described by a set of four coupled first-order ordinary differential equations. This has been investigated extensively using laboratory experiments, Pspice simulation and numerical analysis.

scholarly journals Sumudu transform fundamental properties investigations and applications

2006 ◽  
Vol 2006 ◽  
pp. 1-23 ◽  
Author(s):  
Fethi Bin Muhammed Belgacem ◽  
Ahmed Abdullatif Karaballi
Keyword(s):  
Problem Solving ◽  
Differential Equations ◽  
Laplace Transform ◽  
Frequency Domain ◽  
Paradigm Shift ◽  
Complex Formulation ◽  
Fundamental Properties ◽  
The Laplace Transform ◽  
Sumudu Transform ◽  
Sumudu Transforms

The Sumudu transform, whose fundamental properties are presented in this paper, is still not widely known, nor used. Having scale and unit-preserving properties, the Sumudu transform may be used to solve problems without resorting to a new frequency domain. In 2003, Belgacem et al have shown it to be the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving. Here, using the Laplace-Sumudu duality (LSD), we avail the reader with a complex formulation for the inverse Sumudu transform. Furthermore, we generalize all existing Sumudu differentiation, integration, and convolution theorems in the existing literature. We also generalize all existing Sumudu shifting theorems, and introduce new results and recurrence results, in this regard. Moreover, we use the Sumudu shift theorems to introduce a paradigm shift into the thinking of transform usage, with respect to solving differential equations, that may be unique to this transform due to its unit-preserving properties. Finally, we provide a large and more comprehensive list of Sumudu transforms of functions than is available in the literature.

Factorization Ladders and Eigenfunctions

1949 ◽  
Vol 1 (4) ◽  
pp. 379-396 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Manufacturing Process ◽  
Boundary Value ◽  
Factorization Method ◽  
Sturm Liouville ◽  
The Subject

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

scholarly journals New Numerical Treatment for a Family of Two-Dimensional Fractional Fredholm Integro-Differential Equations

Algorithms ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 37
Author(s):  
Amer Darweesh ◽  
Marwan Alquran ◽  
Khawla Aghzawi
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Haar Wavelet ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Wavelet Method ◽  
New Approach ◽  
Robust Algorithm ◽  
Haar Wavelet Method ◽  
The Laplace Transform

In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.

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