A synthetical development approach for rehabilitation assistive smart product–service systems: A case study

Digital transformation (DT), the combination of information, computing, communication and connectivity technologies, which has triggered an effective upgrade of different aspects of market strategy, customer experience etc. Nowadays, rehabilitation assistive devices (RADs) are evolving to be more digital, intelligent and personalized. Digitalization and servitization have fostered to an emerging business model—the smart product–service system (Smart PSS). Therefore, DT of the RADs’ industry advocates not only the design of products and functions, the more important is the management of service processes and resource integration. With the increase in the elderly and disabled population, the requirement for RADs is becoming more urgent. However, research on Smart PSS for RAD is still limited. The rehabilitation assistive smart product–service systems (RASPSS) was introduced into the development of RADs based on the “Design and Management of DT” strategy through the service design of assistive devices and user requirements analysis. Further, an integrated design of RAD and Smart PSS has been created, a development method of RASPSS proposed, the theoretical model of the Smart PSS based on RADs built. To specify the service framework, this case study discusses the development of a home rehabilitation assistive system for femoral stem fracture patients. This paper evaluates the usability of the system, the results of which prove usability and effectiveness of the RASPSS development method. The RASPSS development model is designed to meet needs of stakeholders, improve the user rehabilitation experience, promote the service innovation of Smart PSS, bring certain market benefits of rehabilitation aids and create social value.     

Dual–primal proximal point algorithms for extended convex programming

This paper presents a class of dual–primal proximal point algorithms (PPAs) for extended convex programming with linear constraints. By choosing appropriate proximal regularization matrices, the application of the general PPA to the equivalent variational inequality of the extended convex programming with linear constraints can result in easy proximal subproblems. In theory, the sequence generated by the general PPA may fail to converge since the proximal regularization matrix is asymmetric sometimes. So we construct descent directions derived from the solution obtained by the general PPA. Different step lengths and descent directions are chosen with the negligible additional computational load. The global convergence of the new algorithms is proved easily based on the fact that the sequences generated are Fejér monotone. Furthermore, we provide a simple proof for the O(1/t) convergence rate of these algorithms.     

A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.     

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.