This research develops a new non-standard scheme based on the Adomian decomposition method (ADM) to solve nonlinear equations. The ADM was adopted to solve the nonlinear differential equation resulting from the discretization of the differential equation. The new scheme does not need to linearize or non-locally linearize the nonlinear term of the differential equation. Two examples are given to demonstrate the efficiency of this scheme.
In this paper, the fully implicit 2-point block backward differentiation formula and diagonally implicit 2-point block
backward differentiation formula were developed under the interpretation of generalized differentiability concept for
solving first order fuzzy differential equations. Some fuzzy initial value problems were tested in order to demonstrate the
performance of the developed methods. The approximated solutions for both methods were in good agreement with the
exact solutions. The numerical results showed that the diagonally implicit method outperforms the fully implicit method
in term of accuracy.
In this paper, the homotopy decomposition method with a modified definition of beta fractional derivative is adopted
to find approximate solutions of higher-dimensional time-fractional diffusion equations. To apply this method, we find
the modified beta integral for both sides of a fractional differential equation first, then using homotopy decomposition
method we can obtain the solution of the integral equation in a series form. We compare the solutions obtained by the
proposed method with the exact solutions obtained using fractional variational homotopy perturbation iteration method
via modified Riemann-Liouville derivative. The comparison shows that the results are in a good agreement.
A third order Nakashima type implicit Pseudo Runge-Kutta method is presented. The free parameter was determined by minimizing the error bound. The stability region of the method was presented. Some problems on delay differential equations are tested to compare the accuracy of the proposed method with third order RADAU I.
Improvements over embedded diagonally implicit Runge-Kutta pair of order four in five are presented. Method of higher stage order with a zero first row and the last row of the coefficient matrix is identical to the vector output is given. The stability aspect of it is also looked into and a standard test problems are solved using the method. Numerical results are tabulated and compared with the existing method.
The improved block hybrid collocation method (KBHK(B)) with two off-step points and four collocation points is proposed
for the direct solution of general third order ordinary differential equations. Modification is done by adding first
derivative of third order function into the general form of block hybrid collocation method to yield KBHK(B). Both main
and additional methods are derived via interpolation (power series function) and collocation of the basic polynomial.
An improved block method is derived to provide the approximation at five points concurrently. Zero stability, consistency,
convergence and absolute stability region of KBHK(B) are investigated. Some numerical examples with exact solution are
tested to illustrate the efficiency of the method. KBHK(B) is compared by other block hibrid collocation method in term
of global error and step number. It is shown that KBHK(B) generates minimum global error with minimum step number.
The Vector Evaluated Particle Swarm Optimisation algorithm is widely used to solve multiobjective optimisation problems. This algorithm optimises one objective using a swarm of particles where their movements are guided by the best solution found by another swarm. However, the best solution of a swarm is only updated when a newly generated solution has better fitness than the best solution at the objective function optimised by that swarm, yielding poor solutions for the multiobjective optimisation problems. Thus, an improved Vector Evaluated Particle Swarm Optimisation algorithm is introduced by incorporating the nondominated solutions as the guidance for a swarm rather than using the best solution from another swarm. In this paper, the performance of improved Vector Evaluated Particle Swarm Optimisation algorithm is investigated using performance measures such as the number of nondominated solutions found, the generational distance, the spread, and the hypervolume. The results suggest that the improved Vector Evaluated Particle Swarm Optimisation algorithm has impressive performance compared with the conventional Vector Evaluated Particle Swarm Optimisation algorithm.
In the original particle swarm optimisation (PSO) algorithm, the particles' velocities and positions are updated after the whole swarm performance is evaluated. This algorithm is also known as synchronous PSO (S-PSO). The strength of this update method is in the exploitation of the information. Asynchronous update PSO (A-PSO) has been proposed as an alternative to S-PSO. A particle in A-PSO updates its velocity and position as soon as its own performance has been evaluated. Hence, particles are updated using partial information, leading to stronger exploration. In this paper, we attempt to improve PSO by merging both update methods to utilise the strengths of both methods. The proposed synchronous-asynchronous PSO (SA-PSO) algorithm divides the particles into smaller groups. The best member of a group and the swarm's best are chosen to lead the search. Members within a group are updated synchronously, while the groups themselves are asynchronously updated. Five well-known unimodal functions, four multimodal functions, and a real world optimisation problem are used to study the performance of SA-PSO, which is compared with the performances of S-PSO and A-PSO. The results are statistically analysed and show that the proposed SA-PSO has performed consistently well.
In this paper, we demonstrate a modified scheme for solving the nonlinear KleinGordon
equation of PDE hyperbolic types. The Klein-Gordon equation is a relativistic
wave equation version of the Schrodinger equation, which is widely used in quantum
mechanics. Additionally, the nonstandard finite difference scheme has been used
extensively to solve differential equations and we have constructed a modified scheme
based on the nonstandard finite difference scheme associated with harmonic mean
averaging for solving the nonlinear inhomogeneous Klein-Gordon equation where the
denominator is replaced by an unusual function. The numerical results obtained have
been compared and showed to have a good agreement with results attained using the
standard finite difference (CTCS) procedure, which provided that the proposed scheme
is reliable. Numerical experiments are tested to validate the accuracy level of the
scheme with the analytical results.
A sequential algorithm is developed for the non-linear dual-sorption model developed by Chandrasekaran et al. [1,2] which monitors pharmacokinetic profiles in percutaneous drug absorption. In the experimental study of percutaneous absorption, it is often observed that the lag-time decreases with the increase in the donor concentration when two or more donor concentrations of the same compound are used. The dual-sorption model has sometimes been employed to explain such experimental results. In this paper, it is shown that another feature observed after vehicle removal may also characterize the dual-sorption model. Soon after vehicle removal, the plots of the drug flux versus time become straight lines on a semilogarithmic scale as in the linear model, but the half-life is prolonged thereafter when the dual-sorption model prevails. The initial half-life after vehicle removal with a low donor concentration is longer than that with a higher donor concentration. These features, if observed in experiments, may be used as evidence to confirm that the dual-sorption model gives an explanation to the non-linear kinetic behaviour of a permeant.
This paper proposes the correction of faulty sensors using a synthesis of the greedy sparse constrained optimization GSCO) technique. The failure of sensors can damage the radiation power pattern in terms of sidelobes and nulls. The synthesis problem can recover the wanted power pattern with reduced number of sensors into the background of greedy algorithm and solved with orthogonal matching pursuit (OMP) technique. Numerical simulation examples of linear arrays are offered to demonstrate the effectiveness of getting the wanted power pattern with a reduced number of antenna sensors which is compared with the available techniques in terms of sidelobes level and number of nulls.
In this paper, extended Runge-Kutta fourth order method for directly solving the fuzzy logistic problem is presented. The extended Runge-Kutta method has lower number of function evaluations, compared with the classical Runge-Kutta method. The numerical robustness of the method in parameter estimation is enhanced via error minimization in predicting growth rate and carrying capacity. The results of fuzzy logistic model with the estimated parameters have been compared with population growth data in Malaysia, which indicate that this method is more accurate that the data population. Numerical example is given to illustrate the efficiency of the proposed model. It is concluded that robust parameter estimation technique is efficient in modelling population growth.
Wireless Body Area Networks (WBANs) are single-hop network systems, where sensors gather the body's vital signs and send them directly to master nodes (MNs). The sensors are distributed in or on the body. Therefore, body posture, clothing, muscle movement, body temperature, and climatic conditions generally influence the quality of the wireless link between sensors and the destination. Hence, in some cases, single hop transmission ('direct transmission') is not sufficient to deliver the signals to the destination. Therefore, we propose an emergency-based cooperative communication protocol for WBAN, named Critical Data-based Incremental Cooperative Communication (CD-ICC), based on the IEEE 802.15.6 CSMA standard but assuming a lognormal shadowing channel model. In this paper, a complete study of a system model is inspected in the terms of the channel path loss, the successful transmission probability, and the outage probability. Then a mathematical model is derived for the proposed protocol, end-to-end delay, duty cycle, and average power consumption. A new back-off time is proposed within CD-ICC, which ensures the best relays cooperate in a distributed manner. The design objective of the CD-ICC is to reduce the end-to-end delay, the duty cycle, and the average power transmission. The simulation and numerical results presented here show that, under general conditions, CD-ICC can enhance network performance compared to direct transmission mode (DTM) IEEE 802.15.6 CSMA and benchmarking. To this end, we have shown that the power saving when using CD-ICC is 37.5% with respect to DTM IEEE 802.15.6 CSMA and 10% with respect to MI-ICC.
Legendre moments are continuous moments, hence, when applied to discrete-space images, numerical approximation is involved and error occurs. This paper proposes a method to compute the exact values of the moments by mathematically integrating the Legendre polynomials over the corresponding intervals of the image pixels. Experimental results show that the values obtained match those calculated theoretically, and the image reconstructed from these moments have lower error than that of the conventional methods for the same order. Although the same set of exact Legendre moments can be obtained indirectly from the set of geometric moments, the computation time taken is much longer than the proposed method.
The simultaneous effect of flexible wall and multiple stenoses on the flow and mass transfer of blood is investigated through numerical computation and simulations. The solution is obtained using the Marker and Cell technique on an axisymmetric model of Newtonian blood flow. The results compare favorably with physical observations where the pulsatile boundary condition and double stenoses result in a higher pressure drop across the stenoses. The streamlines, the iso-concentration lines, the Sherwood number, and the mass concentration variations along the entire wall segment provide a comprehensive analysis of the mass transport characteristics. The double stenoses and pulsatile inlet conditions increase the number of recirculation regions and effect a higher mass transfer rate at the throat, whereby more mass is expected to accumulate and cause further stenosis.
Interference resulting from Cognitive Radios (CRs) is the most important aspect of cognitive radio networks that leads to degradation in Quality of Service (QoS) in both primary and CR systems. Power control is one of the efficient techniques that can be used to reduce interference and satisfy the Signal-to-Interference Ratio (SIR) constraint among CRs. This paper proposes a new distributed power control algorithm based on game theory approach in cognitive radio networks. The proposal focuses on the channel status of cognitive radio users to improve system performance. A new cost function for SIR-based power control via a sigmoid weighting factor is introduced. The existence of Nash Equilibrium and convergence of the algorithm are also proved. The advantage of the proposed algorithm is the possibility to utilize and implement it in a distributed manner. Simulation results show considerable savings on Nash Equilibrium power compared to relevant algorithms while reduction in achieved SIR is insignificant.
We study generalized variable projective synchronization between two unified time delayed systems with constant and modulated time delays. A novel Krasovskii-Lyapunov functional is constructed and a generalized sufficient condition for synchronization is derived analytically using the Lyapunov stability theory and adaptive techniques. The proposed scheme is valid for a system of n-numbers of first order delay differential equations. Finally, a new neural oscillator is considered as a numerical example to show the effectiveness of the proposed scheme.
A one-dimensional biofilm model was developed based on the basic principle of conservation of mass. Three simple, generic processes were combined in the model which includes microbial growth, diffusive and convective mass transport. The final model could generate a quantitative description of the relationship between the microbial growth and the consumption of substrate (oxygen) within the fixed biofilm thickness. Mass transfer resistance contributes large influence on the substrates and microbial concentration across the biofilm thickness due to the effect of biofilm structure.
Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in vivo and in vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue-synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.