We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the k-path Laplacian operators L k , which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the k-path Laplacian operators are self-adjoint. Then, we study the transformed k-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplaceand factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed k-path Laplacians ∞ k=1 k −s L k produces superdiffusive processes when 1 < s < 3.
In this paper we consider a generalized diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian. In particular, we prove that superdiffusion occurs when the parameter s in the Mellin transform is in the interval (2, 4) and that normal diffusion prevails when s > 4.2010 Mathematics Subject Classification. 47B39; 60J60, 05C81.
Image inpainting is the process of recovering the damage areas in the images in an undetectable way, it is considered the important one of the subjects in image processing. There are many applications of image inpainting include the restoration of damaged images, paintings, and movies, to the removal of selected objects, such as text, lines, subtitles, publicity, and stamps. The main objective of inpainting is to reconstruct the missing region in such a way that the observer does not come to know that the image has been manipulated. Inpainting methods can be categorized into global and local methods, the global methods are applied to reconstruct the damaged areas in the image based on the information in the data of images that have the same content. While the local methods are used to reconstruct the missing regions based on the information in the rest parts of the image. There are several local methods proposed for image inpainting such as PDE-based inpainting (PDE-BI), exemplar-based inpainting (EBI), hybrid, and texture synthesis methods. In this paper, a review of different PDE and variational methods used for image inpainting is provided. Different PDE-BI methods like 2nd-and high-order of variational and PDE methods are discussed with its pros and cons.
The modeling of many phenomena in various fields such as mathematics, physics, chemistry, engineering, biology, and astronomy is done by the nonlinear partial differential equations (PDE). The hyperbolic telegraph equation is one of them, where it describes the vibrations of structures (e.g., buildings, beams, and machines) and are the basis for fundamental equations of atomic physics. There are several analytical and numerical methods are used to solve the telegraph equation. An analytical solution considers framing the problem in a well-understood form and calculating the exact resolution. It also helps to understand the answers to the problem in terms of accuracy and convergence. These analytic methods have limitations with accuracy and convergence. Therefore, a novel analytic approximate method is proposed to deal with constraints in this paper. This method uses the Taylors' series in its derivation. The proposed method has used for solving the secondorder, hyperbolic equation (Telegraph equation) with the initial condition. Three examples have presented to check the effectiveness, accuracy, and convergence of the method. The solutions of the proposed method also compared with those obtained by the Adomian decomposition method (ADM), and the Homotopy analysis method (HAM). The technique is easy to implement and produces accurate results. In particular, these results display that the proposed method is efficient and better than the other methods in terms of accuracy and convergence.
In this paper, an extension of the idea of the best approximation in the Hölder spaces with respect to Fourier-Jacobi operators by moduli of smoothness is studied. A special form of the moduli of smoothness is considered to get a strong convergence. Further, advanced approaches of approximation and some direct and inverse results are proved. Moreover, the Jackson-type estimate of functions in Hölder spaces by Jacobi transformations to algebraic polynomials with generalized de la Vallée Poussin mean are established.
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