Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSolving PDEs of fractional order using the unified transform method
123
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem.
rate research
Read More
In this paper, an Artificial Neural Network (ANN) technique is developed to find solution of celebrated Fractional order Differential Equations (FDE). Compared to integer order differential equation, FDE has the advantage that it can better describe sometimes various real world application problems of physical systems. Here we have employed multi-layer feed forward neural architecture and error back propagation algorithm with unsupervised learning for minimizing the error function and modification of the parameters (weights and biases). Combining the initial conditions with the ANN output gives us a suitable approximate solution of FDE. To prove the applicability of the concept, some illustrative examples are provided to demonstrate the precision and effectiveness of this method. Comparison of the present results with other available results by traditional methods shows a close match which establishes its correctness and accuracy of this method.
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations.
We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently developed quadrature formula unique to the localized bases we consider. We present error estimates and investigate the stability of the discrete stiffness matrix. Implementation and numerical experiments are discussed.
In these notes we will present (a part of) the parabolic tent spaces theory and then apply it in solving some PDEs originated from the fluid mechanics. In more details, to our most interest are the incompressible homogeneous Navier-Stokes equations. These equations have been investigated mathematically for almost one century. Yet, the question of proving well-posedness (i.e. existence, uniqueness and regularity of solutions) lacks satisfactory answer. A large part of the known positive results in connection with Navier-Stokes equations are those in which the initial data $u_0$ is supposed to have a small norm in some critical or scaling invariant functional space. All those spaces are embedded in the homogeneous Besov space $dot B^{-1}_{infty,infty}$. A breakthrough was made in the paper [16] by Koch and Tataru, where the authors showed the existence and the uniqueness of solutions to the Navier-Stokes system in case when the norm $|u_0|_{mathrm{BMO}^{-1}}$ is small enough. The principal goal of these notes is to present a new proof of the theorem by Koch and Tataru on the Navier-Stokes system, namely the one using the tent spaces theory. We also hope that after having read these notes, the reader will be convinced that the theory of tent spaces is highly likely to be useful in the study of other equations in fluid mechanics. These notes are mainly based on the content of the article [1] by P. Auscher and D. Frey. However, in [1] the authors deal with a slightly more general system of parabolic equations of Navier-Stokes type. Here we have chosen to write down a self-contained text treating only the relatively easier case of the classical incompressible homogeneous Navier-Stokes equations.
We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poissons equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
