The shifted Jacobi polynomial integral operational matrix for solving Riccati differential equation of fractional order

In this article, we have applied Jacobi polynomial to solve Riccati differential equation of fractional order. To do so, we have presented a general formula for the Jacobi operational matrix of fractional integral operator. Using the Tau method, the solution of this problem reduces to the solution of a system of algebraic equations. The numerical results for the examples presented in this paper demonstrate the efficiency of the present method

Photocatalytic Nanolithography of Self-Assembled Monolayers and Proteins

Self-assembled monolayers of alkylthiolates on gold and alkylsilanes on silicon dioxide have been patterned photocatalytically on sub-100 nm length-scales using both apertured near-field and apertureless methods. Apertured lithography was carried out by means of an argon ion laser (364 nm) coupled to cantilever-type near-field probes with a thin film of titania deposited over the aperture. Apertureless lithography was carried out with a helium–cadmium laser (325 nm) to excite titanium-coated, contact-mode atomic force microscope (AFM) probes. This latter approach is readily implementable on any commercial AFM system. Photodegradation occurred in both cases through the localized photocatalytic degradation of the monolayer. For alkanethiols, degradation of one thiol exposed the bare substrate, enabling refunctionalization of the bare gold by a second, contrasting thiol. For alkylsilanes, degradation of the adsorbate molecule provided a facile means for protein patterning. Lines were written in a protein-resistant film formed by the adsorption of oligo(ethylene glycol)-functionalized trichlorosilanes on glass, leading to the formation of sub-100 nm adhesive, aldehyde-functionalized regions. These were derivatized with aminobutylnitrilotriacetic acid, and complexed with Ni2+, enabling the binding of histidine-labeled green fluorescent protein, which yielded bright fluorescence from 70-nm-wide lines that could be imaged clearly in a confocal microscope

Achievable tolerances in robotic feature machining operations using a low-cost hexapod

Portable robotic machine tools potentially allow feature machining processes to be brought to large parts in various industries, creating an opportunity for capital expenditure and operating cost reduction. However, robots lack the machining capability of conventional equipment, which ultimately results in dimensional errors in parts. This work showcases a low-cost hexapod-based robotic machine tool and presents experimental research conducted to investigate how the widely researched robotic machining challenges, e.g. structural dynamics and kinematics, translate to achievable tolerance ranges in real-world production to highlight currently feasible applications and provide a context for considering technology improvements. Machining trials assess the total dimensional errors in the final part over multiple geometries. A key finding is error variation which is in the sub-millimetre range, although, in some cases, upper tolerance limits < 100 μm are achieved. Practical challenges are also noted. Most significantly, it is demonstrated that dimensional machining error is mainly systematic in nature and therefore that the total error can be dramatically reduced with in situ measurement and compensation. Potential is therefore found to achieve a flexible, high-performance robotic machining capability despite complex and diverse underlying scientific challenges. Overall, the work presented highlights achievable tolerances in low-cost robotic machining and opportunities for improvement, also providing a practical benchmark useful for process selection

A method for solving nonlinear Volterra’s population growth model of noninteger order

Abstract Many numerical methods have been developed for nonlinear fractional integro-differential Volterra’s population model (FVPG). In these methods, to approximate a function on a particular interval, only a restricted number of points have been employed. In this research, we show that it is possible to use the fuzzy transform method (F-transform) to tackle with FVPG. It makes the F-transform preferable to other methods since it can make full use of all points on this interval. We also make a comparison showing that this method is less computational and is more convenient to be utilized for coping with nonlinear integro-differential equation (IDEs), fractional nonlinear integro-differential equation (FIDEs), and fractional ordinary differential equations (FODEs)