Now showing items 1-20 of 420

• Data-driven selection and parameter estimation for DNA methylation mathematical models

(Elsevier, 2019-01-10)
Epigenetics is coming to the fore as a key process which underpins health. In particular emerging experimental evidence has associated alterations to DNA methylation status with healthspan and aging. Mammalian DNA methylation status is maintained by an intricate array of biochemical and molecular processes. It can be argued changes to these fundamental cellular processes ultimately drive the formation of aberrant DNA methylation patterns, which are a hallmark of diseases, such as cancer, Alzheimer's disease and cardiovascular disease. In recent years mathematical models have been used as e ective tools to help advance our understanding of the dynamics which underpin DNA methylation. In this paper we present linear and nonlinear models which encapsulate the dynamics of the molecular mechanisms which de ne DNA methylation. Applying a recently developed Bayesian algorithm for parameter estimation and model selection, we are able to estimate distributions of parameters which include nominal parameter values. Using limited noisy observations, the method also identifed which methylation model the observations originated from, signaling that our method has practical applications in identifying what models best match the biological data for DNA methylation.
• Data-driven selection and parameter estimation for DNA methylation mathematical models.

(2019-01-08)
Epigenetics is coming to the fore as a key process which underpins health. In particular emerging experimental evidence has associated alterations to DNA methylation status with healthspan and aging. Mammalian DNA methylation status is maintained by an intricate array of biochemical and molecular processes. It can be argued changes to these fundamental cellular processes ultimately drive the formation of aberrant DNA methylation patterns, which are a hallmark of diseases, such as cancer, Alzheimer's disease and cardiovascular disease. In recent years mathematical models have been used as effective tools to help advance our understanding of the dynamics which underpin DNA methylation. In this paper we present linear and nonlinear models which encapsulate the dynamics of the molecular mechanisms which define DNA methylation. Applying a recently developed Bayesian algorithm for parameter estimation and model selection, we are able to estimate distributions of parameters which include nominal parameter values. Using limited noisy observations, the method also identified which methylation model the observations originated from, signaling that our method has practical applications in identifying what models best match the biological data for DNA methylation. [Abstract copyright: Copyright © 2019. Published by Elsevier Ltd.]
• Appearance Modeling of Living Human Tissues

(Wiley, 2019)
The visual fidelity of realistic renderings in Computer Graphics depends fundamentally upon how we model the appearance of objects resulting from the interaction between light and matter reaching the eye. In this paper, we survey the research addressing appearance modeling of living human tissue. Among the many classes of natural materials already researched in Computer Graphics, living human tissues such as blood and skin have recently seen an increase in attention from graphics research. There is already an incipient but substantial body of literature on this topic, but we also lack a structured review as presented here. We introduce a classification for the approaches using the four types of human tissues as classifiers. We show a growing trend of solutions that use first principles from Physics and Biology as fundamental knowledge upon which the models are built. The organic quality of visual results provided by these Biophysical approaches is mainly determined by the optical properties of biophysical components interacting with light. Beyond just picture making, these models can be used in predictive simulations, with the potential for impact in many other areas.
• Bordered Constructions of Self-Dual Codes from Group Rings and New Extremal Binary Self-Dual Codes

(Elsevier, 2019)
We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring $\F_2+u\F_2$, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.
• Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

(Elsevier, 2018-12-17)
We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• Towards Organisational Learning Enhancement: Assessing Software Engineering Practice

(Emerald Publishing Limited, 2018-12-17)
• Purpose – Issues surrounding knowledge management, knowledge transfer and learning within organisations challenge continuity and resilience in the face of changing environments. While initiatives are principally applied within large organisations, there is scope to assess how the processes are handled within small and medium enterprises (SMEs) and to consider how they might be enhanced. This paper presents an evaluation of practice within an evolving software development unit to determine what has been learned and how the knowledge acquired has been utilised to further organisational development. These results provide the basis for the design and implementation of a proposed support tool to enhance professional practice. • Design/methodology/approach – A small software development unit, which has successfully delivered bespoke systems since its establishment a number of years ago, was selected for analysis. The unit operates as a team whose actions and behaviours were identified and validated by the following means: in-depth interviews were carried out with each member of the team to elicit an understanding of individual and collective development. Interview data were recorded and transcribed and subjected to qualitative analysis to identify key themes underpinning knowledge acquisition and utilisation. Samples of project documentation were scrutinised to corroborate interview data. After analysing the data, a focus-group meeting was held to validate the results and to generate further insights into learning within the team. • Findings - Qualitative analysis of the data revealed key changes in thinking and practice within the team as well as insight into the development of individual and collective contextual knowledge, tacit understanding and learning. This analysis informed the proposal of a bespoke, lightweight, web-based system to support knowledge capture and organisational learning (OL). This approach has the potential to promote resilience and to enhance practice in similar small or start-up enterprises. • Research limitations/implications – Purposeful sampling was used in selecting a small software development team. This enabled in-depth interviewing of all members of the team. This offered a rich environment from which to derive awareness and understanding of individual and collective knowledge acquisition and learning. Focusing on a single small enterprise limits the extent to which the findings can be generalised. However, the research provides evidence of effective practice and learning and has identified themes for the development of a support tool. This approach can be extended to similar domains to advance research into learning and development. • Practical implications – Results of the work undertaken so far have generated promising foundations for the proposed support tool. This offers software developers a system within which they can reflect upon, and record, key learning events affecting technical, managerial and professional practice. • Originality/value – Small enterprises have limited resources to support OL. The qualitative research undertaken so far has yielded valuable insight into the successful development of a single software development team. The construction of a support tool to enhance knowledge acquisition and learning has the capacity to consolidate valuable, and potentially scarce, expertise. It also has the potential to facilitate further research to determine how the prototype might be extended or revised to improve its contribution to the team’s development.
• Rapid, Chemical-Free Generation of Optically Scattering Structures in Poly(ethylene terephthalate) Using a CO2 Laser for Lightweight and Flexible Photovoltaic Applications

(Hindawi, 2018-12-16)
Highly light scattering structures have been generated in a poly(ethylene terephthalate) (PET) film using a CO2 laser. The haze, and in some cases the transparency, of the PET films have been improved by varying the processing parameters of the laser (namely, scanning velocity, laser output power, and spacing between processed tracks). When compared with the unprocessed PET, the haze has improved from an average value of 3.26% to a peak of 55.42%, which equates to an absolute improvement of 52.16% or a 17-fold increase. In addition to the optical properties, the surfaces have been characterised using optical microscopy and mapped with an optical profilometer. Key surface parameters that equate to the amount and structure of surface roughness and features have been analysed. The CO2 laser generates microstructures at high speed, without affecting the bulk properties of the material, and is inherently a chemical-free process making it particularly applicable for use in industry, fitting well with the high-throughput, roll to roll processes associated with the production of flexible organic photovoltaic devices.
• Numerical Solution of Fractional Differential Equations and their Application to Physics and Engineering

(University of Chester, 2018-12-03)
This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²<sup>-α</sup>), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.
• Enhancement in Interfacial Adhesion of Ti/Polyetheretherketone by Electrophoretic Deposition of Graphene Oxide

(Wiley, 2018-11-25)
• Controller Design Methodology for Sustainable Local Energy Systems

(University of Chester, 2018-11-15)
Commercial Buildings and complexes are no longer just national heat and power network energy loads, but they are becoming part of a smarter grid by including their own dedicated local heat and power generation. They do this by utilising both heat and power networks/micro-grids. A building integrated approach of Combined Heat and Power (CHP) generation with photovoltaic power generation (PV) abbreviated as CHPV is emerging as a complementary energy supply solution to conventional (i.e. national grid based) gas and electricity grid supplies in the design of sustainable commercial buildings and communities. The merits for the building user/owner of this approach are: to reduce life time energy running costs; reduce carbon emissions to contribute to UK’s 2020/2030 climate change targets; and provide a more flexible and controllable local energy system to act as a dynamic supply and/or load to the central grid infrastructure. The energy efficiency and carbon dioxide (CO2) reductions achievable by CHP systems are well documented. The merits claimed by these solutions are predicated on the ability of these systems being able to satisfy: perfect matching of heat and power supply and demand; ability at all times to maintain high quality power supply; and to be able to operate with these constraints in a highly dynamic and unpredictable heat and power demand situation. Any circumstance resulting in failure to guarantee power quality or matching of supply and demand will result in a degradation of the achievable energy efficiency and CO2 reduction. CHP based local energy systems cannot rely on large scale diversity of demand to create a relatively easy approach to supply and demand matching (i.e. as in the case of large centralised power grid infrastructures). The diversity of demand in a local energy system is both much greater than the centralised system and is also specific to the local system. It is therefore essential that these systems have robust and high performance control systems to ensure supply and demand matching and high power quality can be achieved at all times. Ideally this same control system should be able to make best use of local energy system energy storage to enable it to be used as a flexible, highly responsive energy supply and/or demand for the centralised infrastructure. In this thesis, a comprehensive literature survey has identified that there is no scientific and rigorous method to assess the controllability or the design of control systems for these local energy systems. Thus, the main challenge of the work described in this thesis is that of a controller design method and modelling approach for CHP based local energy systems. Specifically, the main research challenge for the controller design and modelling methodology was to provide an accurate and stable system performance to deliver a reliable tracking of power drawn/supplied to the centralised infrastructure whilst tracking the require thermal comfort in the local energy systems buildings. In the thesis, the CHPV system has been used as a case study. A CHPV based solution provides all the benefits of CHP combined with the near zero carbon building/local network integrated PV power generation. CHPV needs to be designed to provide energy for the local buildings’ heating, dynamic ventilating system and air-conditioning (HVAC) facilities as well as all electrical power demands. The thesis also presents in addition to the controller design and modelling methodology a novel CHPV system design topology for robust, reliable and high-performance control of building temperatures and energy supply from the local energy system. The advanced control system solution aims to achieve desired building temperatures using thermostatic control whilst simultaneously tracking a specified national grid power demand profile. The theory is innovative as it provides a stability criterion as well as guarantees to track a specified dynamic grid connection demand profile. This research also presents: design a dynamic MATLAB simulation model for a 5-building zone commercial building to show the efficacy of the novel control strategy in terms of: delivering accurate thermal comfort and power supply; reducing the amount of CO2 emissions by the entire energy system; reducing running costs verses national rid/conventional approaches. The model was developed by inspecting the functional needs of 3 local energy system case studies which are also described in the thesis. The CHPV system is combined with supplementary gas boiler for additional heating to guarantee simultaneous tracking of all the zones thermal comfort requirements whilst simultaneously tracking a specified national grid power demand using a Photovoltaics array to supply the system with renewable energy to reduce amount of CO2 emission. The local energy system in this research can operate in any of three modes (Exporting, Importing, Island). The emphasise of the thesis modelling method has been verified to be applicable to a wide range of case studies described in the thesis chapter 3. This modelling framework is the platform for creating a generic controlled design methodology that can be applied to all these case studies and beyond, including Local Energy System (LES) in hotter climates that require a cooling network using absorption chillers. In the thesis in chapter 4 this controller design methodology using the modelling framework is applied to just one case study of Copperas Hill. Local energy systems face two types of challenges: technical and nontechnical (such as energy economics and legislation). This thesis concentrates solely on the main technical challenges of a local energy system that has been identified as a gap in knowledge in the literature survey. The gap identified is the need for a controller design methodology to allow high performance and safe integration of the local energy system with the national grid infrastructure and locally installed renewables. This integration requires the system to be able to operate at high performance and safely in all different modes of operation and manage effectively the multi-vector energy supply system (e.g. simultaneous supply of heat and power from a single system).
• Mathematical models of DNA methylation dynamics: Implications for health and ageing.

(2018-11-15)
DNA methylation is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimer's disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fall short of fully capturing the dynamics of this process. In this paper we develop a linear and nonlinear model which captures more fully the dynamics of the key intracellular events which characterise DNA methylation. In particular the outcomes of our linear model result in gene promoter specific methylation levels which are more biologically plausible than those revealed by previous mathematical models. In addition, our nonlinear model predicts DNA methylation promoter bistability which is commonly observed experimentally. The findings from our models have implications for our current understanding of how changes to the dynamics which underpin DNA methylation affect ageing and health. We also propose how our ideas can be tested in the lab. [Abstract copyright: Copyright © 2018 Elsevier Ltd. All rights reserved.]
• In-depth synthetic, physicochemical and in vitro biological investigation of a new ternary V(IV) antioxidant material based on curcumin.

(Elsevier, 2018-11-06)
Curcumin is a natural product with a broad spectrum of beneficial properties relating to pharmaceutical applications, extending from traditional remedies to modern cosmetics. The biological activity of such pigments, however, is limited by their solubility and bioavailability, thereby necessitating new ways of achieving optimal tissue cellular response and efficacy as drugs. Metal ion complexation provides a significant route toward improvement of curcumin stability and biological activity, with vanadium being a representative such metal ion, amply encountered in biological systems and exhibiting exogenous bioactivity through potential pharmaceuticals. Driven by the need to optimally increase curcumin bioavailability and bioactivity through complexation, synthetic efforts were launched to seek out stable species, ultimately leading to the synthesis and isolation of a new ternary V(IV)-curcumin-(2,2’-bipyridine) complex. Physicochemical characterization (elemental analysis, FT-IR, Thermogravimetry (TGA), UV-Visible, NMR, ESI-MS, Fluorescence, X-rays) portrayed the solid-state and solution properties of the ternary complex. Pulsed-EPR spectroscopy, in frozen solutions, suggested the presence of two species, cis- and trans-conformers. Density Functional Theory (DFT) calculations revealed the salient features and energetics of the two conformers, thereby complementing EPR spectroscopy. The well-described profile of the vanadium species led to its in vitro biological investigation involving toxicity, cell metabolism inhibition in S. cerevisiae cultures, Reactive Oxygen Species (ROS)-suppressing capacity, lipid peroxidation, and plasmid DNA degradation. A multitude of bio-assays and methodologies, in comparison to free curcumin, showed that it exhibits its antioxidant potential in a concentration-dependent fashion, thereby formulating a bioreactivity profile supporting development of new efficient vanado-pharmaceuticals, targeting (extra)intra-cellular processes under (patho)physiological conditions.
• Disrupting folate metabolism reduces the capacity of bacteria in exponential growth to develop persisters to antibiotics

(Microbiology Society, 2018-11-01)
Bacteria can survive high doses of antibiotics through stochastic phenotypic diversification. We present initial evidence that folate metabolism could be involved with the formation of persisters. The aberrant expression of the folate enzyme gene fau seems to reduce the incidence of persisters to antibiotics. Folate-impaired bacteria had a lower generation rate for persisters to the antibiotics ampicillin and ofloxacin. Persister bacteria were detectable from the outset of the exponential growth phase in the complex media. Gene expression analyses tentatively showed distinctive profiles in exponential growth at times when bacteria persisters were observed. Levels of persisters were assessed in bacteria with altered, genetically and pharmacologically, folate metabolism. This work shows that by disrupting folate biosynthesis and usage, bacterial tolerance to antibiotics seems to be diminished. Based on these findings there is a possibility that bacteriostatic antibiotics such as anti-folates could have a role to play in clinical settings where the incidence of antibiotic persisters seems to drive recalcitrant infections.
• Human adipose tissue-derived mesenchymal stem/stromal cells adhere to and inhibit the growth of Staphylococcus aureus and Pseudomonas aeruginosa.

(2018-10-23)
We have cultured and phenotyped human adipose tissue-derived mesenchymal stem/stromal cells (AT MSCs) and inoculated these cultures with bacteria common to infected skin wounds, i.e. Staphylococcus aureus and Pseudomonas aeruginosa. Cell interactions were examined by scanning electron microscopy (SEM), whilst bacterial growth was measured by colony forming unit (c.f.u.) and biofilm assays. AT MSCs appeared to attach to the bacteria and to engulf S. aureus. Significantly fewer bacterial c.f.u. were present in AT MSC : bacterial co-cultures compared with bacteria cultured alone. Antibacterial activity, including an inhibition of P. aeruginosa biofilm formation, was observed when bacteria were treated with conditioned medium harvested from the AT MSC :  bacterial co-cultures, irrespective of the bacterial species to which the AT MSCs had been exposed to previously. Hence, we have demonstrated that AT MSCs inhibit the growth of two common bacterial species. This was associated with bacterial adhesion, potential engulfment or phagocytosis, and the secretion of antibacterial factors.
• A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

(Elsevier, 2018-10-05)
A new high-order finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k-1}) \big ), k=1, 2, \dots, N,$ with the convergence order $O(\Delta t^{4-\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4-\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• Insights into HOx and ROx chemistry in the boreal forest via measurement of peroxyacetic acid, peroxyacetic nitric anhydride (PAN) and hydrogen peroxide

(European Geosciences Union, 2018-09-21)
• An Information-Theoretic Approach to the Cost-benefit Analysis of Visualization in Virtual Environments

(IEEE, 2018-08-20)
Visualization and virtual environments (VEs) have been two interconnected parallel strands in visual computing for decades. Some VEs have been purposely developed for visualization applications, while many visualization applications are exemplary showcases in general-purpose VEs. Because of the development and operation costs of VEs, the majority of visualization applications in practice have yet to benefit from the capacity of VEs. In this paper, we examine this status quo from an information-theoretic perspective. Our objectives are to conduct cost-benefit analysis on typical VE systems (including augmented and mixed reality, theatre-based systems, and large powerwalls), to explain why some visualization applications benefit more from VEs than others, and to sketch out pathways for the future development of visualization applications in VEs. We support our theoretical propositions and analysis using theories and discoveries in the literature of cognitive sciences and the practical evidence reported in the literatures of visualization and VEs.
• Oxidation processes in the eastern Mediterranean atmosphere: evidence from the modelling of HOx measurements over Cyprus

(Copernicus Publications, 2018-07-31)
The Mediterranean is a climatically sensitive region located at the crossroads of air masses from three continents: Europe, Africa, and Asia. The chemical processing of air masses over this region has implications not only for the air quality but also for the long-range transport of air pollution. To obtain a comprehensive understanding of oxidation processes over the Mediterranean, atmospheric concentrations of the hydroxyl radical (OH) and the hydroperoxyl radical (HO2) were measured during an intensive field campaign (CYprus PHotochemistry EXperiment, CYPHEX-2014) in the northwest of Cyprus in the summer of 2014. Very low local anthropogenic and biogenic emissions around the measurement location provided a vantage point to study the contrasts in atmospheric oxidation pathways under highly processed marine air masses and those influenced by relatively fresh emissions from mainland Europe. The CYPHEX measurements were used to evaluate OH and HO2 simulations using a photochemical box model (CAABA/MECCA) constrained with CYPHEX observations of O3, CO, NOx, hydrocarbons, peroxides, and other major HOx (OH+HO2) sources and sinks in a low-NOx environment (<100pptv of NO). The model simulations for OH agreed to within 10% with in situ OH observations. Model simulations for HO2 agreed to within 17% of the in situ observations. However, the model strongly under-predicted HO2 at high terpene concentrations, this under-prediction reaching up to 38% at the highest terpene levels. Different schemes to improve the agreement between observed and modelled HO2, including changing the rate coefficients for the reactions of terpene-generated peroxy radicals (RO2) with NO and HO2 as well as the autoxidation of terpene-generated RO2 species, are explored in this work. The main source of OH in Cyprus was its primary production from O3 photolysis during the day and HONO photolysis during early morning. Recycling contributed about one-third of the total OH production, and the maximum recycling efficiency was about 0.7. CO, which was the largest OH sink, was also the largest HO2 source. The lowest HOx production and losses occurred when the air masses had higher residence time over the oceans.
• Exploration and Implementation of Augmented Reality for External Beam Radiotherapy

(University of Chester, 2018-07-17)
Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.