Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

Parameter estimation for a phenomenological model of the cardiac action potential

The action potential (ap) of a cardiac cell is made up of a complex balance of ionic currents which flow across the cell membrane in response to electrical excitation of the cell. Biophysically detailed mathematical models of the ap have grown larger in terms of the variables and parameters required to model new findings in subcellular ionic mechanisms. The fitting of parameters to such models has seen a large degree of parameter and module re-use from earlier models. An alternative method for modelling electrically exciteable cardiac tissue is a phenomenological model, which reconstructs tissue level ap wave behaviour without subcellular details. A new parameter estimation technique to fit the morphology of the ap in a four variable phenomenological model is presented. An approximation of a nonlinear ordinary differential equation model is established that corresponds to the given phenomenological model of the cardiac ap. The parameter estimation problem is converted into a minimisation problem for the unknown parameters. A modified hybrid Nelder–Mead simplex search and particle swarm optimization is then used to solve the minimisation problem for the unknown parameters. The successful fitting of data generated from a well known biophysically detailed model is demonstrated. A successful fit to an experimental ap recording that contains both noise and experimental artefacts is also produced. The parameter estimation method’s ability to fit a complex morphology to a model with substantially more parameters than previously used is established.

Mechanical behaviours of pumpkin peel under compression test

Mechanical damages such as bruising, collision and impact during food processing stages diminish quality and quantity of productions as well as efficiency of operations. Studying mechanical characteristics of food materials will help to enhance current industrial practices. Mechanical properties of fruits and vegetables describe how these materials behave under loading in real industrial operations. Optimizing and designing more efficient equipments require accurate and precise information of tissue behaviours. FE modelling of food industrial processes is an effective method of studying interrelation of variables during mechanical operation. In this study, empirical investigation has been done on mechanical properties of pumpkin peel. The test was a part of FE modelling and simulation of mechanical peeling stage of tough skinned vegetables. The compression test has been conducted on Jap variety of pumpkin. Additionally, stress strain curve, bio-yield and toughness of pumpkin skin have been calculated. The required energy for reaching bio-yield point was 493.75, 507.71 and 451.71 N.mm for 1.25, 10 and 20 mm/min loading speed respectively. Average value of force in bio-yield point for pumpkin peel was 310 N.

Capturing affective experiences using the SMS experience sampling (SMS-ES) method

This paper reports the feasibility and methodological considerations of using the Short Message System Experience Sampling (SMS-ES) Method, which is an experience sampling research method developed to assist researchers to collect repeat measures of consumers’ affective experiences. The method combines SMS with web-based technology in a simple yet effective way. It is described using a practical implementation study that collected consumers’ emotions in response to using mobile phones in everyday situations. The method is further evaluated in terms of the quality of data collected in the study, as well as against the methodological considerations for experience sampling studies. These two evaluations suggest that the SMS-ES Method is both a valid and reliable approach for collecting consumers’ affective experiences. Moreover, the method can be applied across a range of for-profit and not-for-profit contexts where researchers want to capture repeated measures of consumers’ affective experiences occurring over a period of time. The benefits of the method are discussed to assist researchers who wish to apply the SMS-ES Method in their own research designs.

Stochastic regularization method for backward Cauchy problem in Banach spaces

We consider a stochastic regularization method for solving the backward Cauchy problem in Banach spaces. An order of convergence is obtained on sourcewise representative elements.

Stochastic analysis of the VEGF receptor response curve

Recently, an analysis of the response curve of the vascular endothelial growth factor (VEGF) receptor and its application to cancer therapy was described in [T. Alarcón, and K. Page, J. R. Soc. Lond. Interface 4, 283–304 (2007)]. The analysis is significantly extended here by demonstrating that an alternative computational strategy, namely the Krylov FSP algorithm for the direct solution of the chemical master equation, is feasible for the study of the receptor model. The new method allows us to further investigate the hypothesis of symmetry in the stochastic fluctuations of the response. Also, by augmenting the original model with a single reversible reaction we formulate a plausible mechanism capable of realizing a bimodal response, which is reported experimentally but which is not exhibited by the original model. The significance of these findings for mechanisms of tumour resistance to antiangiogenic therapy is discussed.