Accurate spectral solutions of first and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method

In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature

New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented

A new MRI rating scale for progressive supranuclear palsy and multiple system atrophy: validity and reliability

AIM To evaluate a standardised MRI acquisition protocol and a new image rating scale for disease severity in patients with progressive supranuclear palsy (PSP) and multiple systems atrophy (MSA) in a large multicentre study. METHODS The MRI protocol consisted of two-dimensional sagittal and axial T1, axial PD, and axial and coronal T2 weighted acquisitions. The 32 item ordinal scale evaluated abnormalities within the basal ganglia and posterior fossa, blind to diagnosis. Among 760 patients in the study population (PSP = 362, MSA = 398), 627 had per protocol images (PSP = 297, MSA = 330). Intra-rater (n = 60) and inter-rater (n = 555) reliability were assessed through Cohen's statistic, and scale structure through principal component analysis (PCA) (n = 441). Internal consistency and reliability were checked. Discriminant and predictive validity of extracted factors and total scores were tested for disease severity as per clinical diagnosis. RESULTS Intra-rater and inter-rater reliability were acceptable for 25 (78%) of the items scored (≥ 0.41). PCA revealed four meaningful clusters of covarying parameters (factor (F) F1: brainstem and cerebellum; F2: midbrain; F3: putamen; F4: other basal ganglia) with good to excellent internal consistency (Cronbach α 0.75-0.93) and moderate to excellent reliability (intraclass coefficient: F1: 0.92; F2: 0.79; F3: 0.71; F4: 0.49). The total score significantly discriminated for disease severity or diagnosis; factorial scores differentially discriminated for disease severity according to diagnosis (PSP: F1-F2; MSA: F2-F3). The total score was significantly related to survival in PSP (p<0.0007) or MSA (p<0.0005), indicating good predictive validity. CONCLUSIONS The scale is suitable for use in the context of multicentre studies and can reliably and consistently measure MRI abnormalities in PSP and MSA. Clinical Trial Registration Number The study protocol was filed in the open clinical trial registry (http://www.clinicaltrials.gov) with ID No NCT00211224

Adapted Shifted ChebyshevU Operational Matrix of Derivatives: Two Algorithms for Solving Even-Order BVPs

Herein, modified orthogonal polynomials are introduced. These polynomials are generated from the second kind of shifted Chebyshev polynomials on the interval [α, β]. The operational matrix of its derivative is constructed. The Tau and Galerkin method with the proposed orthogonal polynomials is used to solve the boundary value problems (BVPs) with even order. The effectiveness of these methods is proved through their application to several BVPs

New spectral second kind Chebyshev wavelets algorithm for solving linear andnonlinear second-order differential equations involving singular and Bratu type equations,” Abstract and Applied Analysis,

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions