Boundary layers in a two-point boundary value problem with a caputo fractional derivative

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Caputo fractional derivative of order ¿ with 1 < ¿ < 2. Writing ¿ for the solution of the problem, it is known that typically ¿¿¿(¿) blows up as ¿ ¿ 0. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: ¿ may exhibit a boundary layer at ¿ = 1 when ¿ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for ¿). This analysis proves that usually no boundary layer can occur in the solution ¿ at ¿ = 0, and that the quantity ¿ = max¿¿[0,1] ¿(¿), where ¿ is the coefficient of the first-order term in the differential operator, is critical: when¿ < 1,noboundarylayerispresentwhen¿isnear1,butwhen¿ = 1thenaboundarylayerat¿ = 1 is possible. Numerical results illustrate the sharpness of most of our results

Good (and Not So Good) practices in computational methods for fractional calculus

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results

Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation

A reaction-diffusion problem with a Caputo time derivative of order = 2 (0; 1) is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time t = 0, and sharp point wise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis

Why fractional derivatives with nonsingular kernels should not be used

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new

Segmentation as a Method for Improving Model Generated Estimates of Recreational Boating Use

The primary purpose of this study is to develop a system of models based on boat storage segments that can generate different estimates of boating use for regions and counties. It is based on the premise that models that incorporate market segmentation are more efficient and generate more accurate estimates of recreational use when a market is comprised of identifiable market segments. The results show that incorporating storage segments as part of the system of models improves estimates of both the amount and spatial distribution of boating use. The boating use estimates generated by the system of models captures the predominate spatial patterns that characterize boating use in Michigan. Model generated estimates of boating use in regions of the state are within 10% of direct survey based estimates for most regions. Model based estimates of both the number of boats in different types of storage and number of boating days are reasonably close to survey estimates for counties where there was an adequate number (30) of questionnaires returned

Segmentation as a Method for Improving Model Generated Estimates of Recreational Boating Use

The primary purpose of this study is to develop a system of models based on boat storage segments that can generate different estimates of boating use for regions and counties. It is based on the premise that models. that incorporate market segmentation are more efficient and generate more accurate estimates of recreational use when a market is comprised of identifiable market segments. The results show that incorporating storage segments as part of the system of models improves estimates of both the amount and spatial distribution of boating use. The boating use estimates generated by the system of models captures the predominate spatial patterns that characterize boating use in Michigan. Model generated estimates of boating use in regions of the state are within 10% of direct survey based estimates for most regions. Model based estimates of both the number of boats in different types of storage and number of boating days are reasonably close to survey estimates for counties where there was an adequate number (30) of questionnaires returned

Clinical diagnostic model for sciatica developed in primary care patients with low back-related leg pain

Background Identification of sciatica may assist timely management but can be challenging in clinical practice. Diagnostic models to identify sciatica have mainly been developed in secondary care settings with conflicting reference standard selection. This study explores the challenges of reference standard selection and aims to ascertain which combination of clinical assessment items best identify sciatica in people seeking primary healthcare. Methods Data on 394 low back-related leg pain consulters were analysed. Potential sciatica indicators were seven clinical assessment items. Two reference standards were used: (i) high confidence sciatica clinical diagnosis; (ii) high confidence sciatica clinical diagnosis with confirmatory magnetic resonance imaging findings. Multivariable logistic regression models were produced for both reference standards. A tool predicting sciatica diagnosis in low back-related leg pain was derived. Latent class modelling explored the validity of the reference standard. Results Model (i) retained five items; model (ii) retained six items. Four items remained in both models: below knee pain, leg pain worse than back pain, positive neural tension tests and neurological deficit. Model (i) was well calibrated (p = 0.18), discrimination was area under the receiver operating characteristic curve (AUC) 0.95 (95% CI 0.93, 0.98). Model (ii) showed good discrimination (AUC 0.82; 0.78, 0.86) but poor calibration (p = 0.004). Bootstrapping revealed minimal overfitting in both models. Agreement between the two latent classes and clinical diagnosis groups defined by model (i) was substantial, and fair for model (ii). Conclusion Four clinical assessment items were common in both reference standard definitions of sciatica. A simple scoring tool for identifying sciatica was developed. These criteria could be used clinically and in research to improve accuracy of identification of this subgroup of back pain patients

Novel approach to characterising individuals with low back-related leg pain: cluster identification with latent class analysis and 12-month follow-up

Traditionally, low back-related leg pain (LBLP) is diagnosed clinically as referred leg pain or sciatica (nerve root involvement). However, within the spectrum of LBLP, we hypothesised that there may be other unrecognised patient subgroups. This study aimed to identify clusters of patients with LBLP using latent class analysis and describe their clinical course. The study population was 609 LBLP primary care consulters. Variables from clinical assessment were included in the latent class analysis. Characteristics of the statistically identified clusters were compared, and their clinical course over 1 year was described. A 5 cluster solution was optimal. Cluster 1 (n = 104) had mild leg pain severity and was considered to represent a referred leg pain group with no clinical signs, suggesting nerve root involvement (sciatica). Cluster 2 (n = 122), cluster 3 (n = 188), and cluster 4 (n = 69) had mild, moderate, and severe pain and disability, respectively, and response to clinical assessment items suggested categories of mild, moderate, and severe sciatica. Cluster 5 (n = 126) had high pain and disability, longer pain duration, and more comorbidities and was difficult to map to a clinical diagnosis. Most improvement for pain and disability was seen in the first 4 months for all clusters. At 12 months, the proportion of patients reporting recovery ranged from 27% for cluster 5 to 45% for cluster 2 (mild sciatica). This is the first study that empirically shows the variability in profile and clinical course of patients with LBLP including sciatica. More homogenous groups were identified, which could be considered in future clinical and research settings