A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

Iterative Technology in a Singular Fractional Boundary Value Problem with q -Difference

In this paper, we apply the iterative technology to establish the existence of solutions for a fractional boundary value problem with q-difference. Explicit iterative sequences are given to approxinate the solutions and the error estimations are also given.

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory（history-dependent,nonlocal,or hereditary） phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

Diffusion tensor imaging of the hippocampus reflects the severity of hippocampal injury induced by global cerebral ischemia/reperfusion injury

At present,predicting the severity of brain injury caused by global cerebral ischemia/reperfusion injury(GCI/RI)is a clinical problem.After such an injury,clinical indicators that can directly reflect neurological dysfunction are lacking.The change in hippocampal microstructure is the key to memory formation and consolidation.Diffusion tensor imaging is a highly sensitive tool for visualizing injury to hippocampal microstructure.Although hippocampal microstructure,brain-derived neurotrophic factor(BDNF),and tropomyosin-related kinase B(Trk B)levels are closely related to nerve injury and the repair process after GCI/RI,whether these indicators can reflect the severity of such hippocampal injury remains unknown.To address this issue,we established rat models of GCI/RI using the four-vessel occlusion method.Diffusion tensor imaging parameters,BDNF,and Trk B levels were correlated with modified neurological severity scores.The results revealed that after GCI/RI,while neurological function was not related to BDNF and Trk B levels,it was related to hippocampal fractional anisotropy.These findings suggest that hippocampal fractional anisotropy can reflect the severity of hippocampal injury after global GCI/RI.The study was approved by the Institutional Animal Care and Use Committee of Capital Medical University,China(approval No.AEEI-2015-139)on November 9,2015.

On analytical solution of system of nonlinear fractional boundary value problems associated with obstacle

In this paper,we use the modified variation of parameters method(MVPM),an elegant coupling of variation of parameters method(VPM)and Adomian’s decomposition method(ADM),for finding the analytical solution of system of nonlinear fractional boundary value problems associated with obstacle.Caputo sense of fractional derivative is used to coup up with fractional term.The results are calculated in terms of series with easily computable components.The used technique is quite easy and convenient for such type problems because it has been previously applied over several nonlinear obstacle systems.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO FRACTIONAL BOUNDARY VALUE PROBLEMS

In this paper, using fixed point theorems of general β-concave operators in ordered Banach space, we obtain the existence and uniqueness of positive solutions to a class of fractional boundary problem. In the end, an example is given to illustrate our main result.

SOLVABILITY FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS AT RESONANCE

The paper deals a fractional functional boundary value problems with integral boundary conditions. Besed on the coincidence degree theory, some existence criteria of solutions at resonance are established.