scholarly journals Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jian Wang ◽  
Kamran ◽  
Ayesha Jamal ◽  
Xuemei Li
Keyword(s):  
Laplace Transform ◽  
Integrodifferential Equation ◽  
Numerical Scheme ◽  
High Accuracy ◽  
Trapezoidal Rule ◽  
Inverse Laplace Transform ◽  
Fredholm Type ◽  
The Laplace Transform ◽  
Linear And Nonlinear ◽  
The Given

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

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.

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

1975 ◽  
Vol 65 (4) ◽  
pp. 927-935
Author(s):  
I. M. Longman ◽  
T. Beer
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
High Accuracy ◽  
Time Variable ◽  
Inversion Method ◽  
The Laplace Transform ◽  
Previously Treated ◽  
The Mean ◽  
Function Approximations ◽  
Laplace Transform Inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

Generalized coupled transient thermoelastic problem of multilayered spheres by the hybrid numerical method

2003 ◽  
Vol 217 (12) ◽  
pp. 1315-1323
Author(s):  
Z Y Lee ◽  
C L Chang
Keyword(s):  
Laplace Transform ◽  
Finite Difference Methods ◽  
Generalized Thermoelasticity ◽  
Inverse Laplace Transform ◽  
Hybrid Numerical Method ◽  
The Matrix ◽  
The Laplace Transform ◽  
Computational Procedures ◽  
Thermoelastic Problems ◽  
Matrix Similarity

This paper deals with axisymmetric quasi-static coupled thermoelastic problems for multilayered spheres. Laplace transforms and finite difference methods are used to analyse the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. Solutions are obtained for the temperature and thermal deformation distributions for the transient and steady state. It is demonstrated that the computational procedures established in this paper are capable of solving the generalized thermoelasticity problem of multilayered spheres.

Analytical solutions of citrate–phosphate coupled model of rice (Oryza sativa L.) roots

2020 ◽  
Vol 13 (07) ◽  
pp. 2050061
Author(s):  
Huiping Zhang ◽  
Shuyue Wang ◽  
Zhonghui Ou
Keyword(s):  
Oryza Sativa ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Oryza Sativa L ◽  
Coupled Model ◽  
Inverse Laplace Transform ◽  
Citrate Solution ◽  
Separation Variable ◽  
The Laplace Transform ◽  
Citrate Phosphate

The citrate secreted by the rice (Oryza sativa L.) roots will promote the absorption of phosphate, and this process is described by the Kirk model. In our work, the Kirk model is divided into citrate sub-model and phosphate sub-model. In the citrate sub-model, we obtain the analytical solution of citrate with the Laplace transform, inverse Laplace transform and convolution theorem. The citrate solution is substituted into the phosphate sub-model, and the analytical solution of phosphate is obtained by the separation variable method. The existence of the solutions can be proved by the comparison test, the Weierstrass M-test and the Abel discriminating method.

The Inverse Laplace Transform of the Product of Two Whittaker Functions

1962 ◽  
Vol 58 (4) ◽  
pp. 580-582 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Laplace Transform ◽  
Whittaker Function ◽  
Inverse Laplace Transform ◽  
Whittaker Functions ◽  
Original Function ◽  
The Laplace Transform ◽  
Image Position

The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters

New identities for the Wright and the Mittag-Leffler functions using the Laplace transform

2014 ◽  
Vol 07 (03) ◽  
pp. 1450038 ◽  
Author(s):  
Alireza Ansari ◽  
Amirhossein Refahi Sheikhani
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Integral Identities

In this paper, we state three theorems for the inverse Laplace transform and using these theorems we obtain new integral identities involving the products of the Wright and Mittag-Leffler functions. The relationships of these integral identities with the Stieltjes transform are also given.

scholarly journals A Synchronization Criterion for Two Hindmarsh-Rose Neurons with Linear and Nonlinear Coupling Functions Based on the Laplace Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Chunlin Su ◽  
Bin Zhen ◽  
Zigen Song
Keyword(s):  
Laplace Transform ◽  
Chaotic Behavior ◽  
Integral Form ◽  
Nonlinear Coupling ◽  
Laplace Transform Method ◽  
Transform Method ◽  
The Laplace Transform ◽  
Analytical Criterion ◽  
Linear And Nonlinear ◽  
The Stability

In this paper, an analytical criterion is proposed to investigate the synchronization between two Hindmarsh-Rose neurons with linear and nonlinear coupling functions based on the Laplace transform method. Different from previous works, the synchronization error system is expressed in its integral form, which is more convenient to analyze. The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation. Then, an analytical criterion for synchronization between the two HR neurons can be given by using the Routh-Hurwitz criterion. Numerical simulations show that the synchronization criterion derived in this paper is valid, regardless of the periodic spikes or burst-spike chaotic behavior of the two HR neurons. Furthermore, the analytical results have almost the same accuracy as the conditional Lyapunov method. In addition, the calculation quantities always are small no matter the linear and nonlinear coupling functions, which show that the approach presented in this paper is easy to be developed to study synchronization between a large number of HR neurons.

scholarly journals The Numerical Solutions of Nonlinear Time-Fractional Differential Equations by LMADM

2021 ◽  
pp. 17-26
Author(s):  
Hameeda Oda AL-Humedi ◽  
Faeza Lafta Hasan
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Small Frequency ◽  
Adomian Method ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

This paper presents a numerical scheme for solving nonlinear time-fractional differential equations in the sense of Caputo. This method relies on the Laplace transform together with the modified Adomian method (LMADM), compared with the Laplace transform combined with the standard Adomian Method (LADM). Furthermore, for the comparison purpose, we applied LMADM and LADM for solving nonlinear time-fractional differential equations to identify the differences and similarities. Finally, we provided two examples regarding the nonlinear time-fractional differential equations, which showed that the convergence of the current scheme results in high accuracy and small frequency to solve this type of equations.

scholarly journals Complete Response of Linear Circuits to Periodic Nonsinusoidal Input in MATLAB

2020 ◽  
Vol 5 (3) ◽  
pp. 70-74
Author(s):  
Iveta Tomčíková
Keyword(s):  
Laplace Transform ◽  
Complete Response ◽  
Complex Frequency ◽  
Analysis Technique ◽  
Periodic Input ◽  
The Laplace Transform ◽  
Linear Circuits ◽  
The Given

The paper deals with the proposal for finding<br />the complete response of dynamic linear circuits to a<br />periodic nonsinusoidal input in the MATLAB environment.<br />A very powerful tool for solving the given problem is to<br />transform the circuits directly into the complex frequency<br />domain using the Laplace transform and then apply the<br />sparse tableau analysis technique to solve them. Applying<br />above-mentioned methods in the MATLAB environment, it<br />is not difficult to find the complete response of dynamic<br />linear circuits to the periodic input.

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