This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one dimension. The construction are based on a newly developed notion weak derivatives, which natural generalizations the well-established integer order theory. In particular, two one-sided introduced analyzed, they reveal more insights about another family so-called symmetric spaces. Many key t...

We study the mathematical and numerical aspects of the estimation of the 3-D shape of a Lambertian scene seen under diffuse illumination. This problem is known as “shape from ambient shading” (SFAS), and its solution consists of integrating a strongly non-local and non-linear Integro-Partial Differential Equation (I-PDE). We provide a first analysis of this global I-PDE, whereas previous work h...

Abstract We consider an optimal control problem containing a system described by partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to integral cost. investigate existence of solutions for such problem. In our study we use Filippov’s approach combined lower closure theorem orientor fields.

Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which essential for deepening our understanding physical processes predicting future responses. However, the PDEs many real-world problems uncertain, calls PDE discovery. We propose symbolic genetic algorithm (SGA-PDE) to discover open-form directly from data without prior knowledge about e...

In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...

Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We foc...