Search Results (88)

This study introduces a physics-based inverse rendering method for determining the reduced scattering and absorption coefficients of turbid materials with arbitrary shapes, using a single image as input. The approach enables fully spectrally-resolved reconstruction of the wavelength-dependent behaviour of the optical properties while also circumventing the specialised sample preparation required by established measurement techniques. Our approach employs a numerical solution of the Radiative Transfer Equation based on an inverse Monte Carlo framework, utilising an improved Levenberg–Marquardt algorithm. By rendering the edge effects accurately, particularly translucency, it becomes possible to differentiate between scattering and absorption from just one image. Importantly, the errors induced by only approximate prior knowledge of the phase function and refractive index of the material were quantified. The method was validated through theoretical studies on three materials spanning a range of optical parameters, initially using a simple cube geometry and later extended to more complex shapes. Evaluated via the CIE $\Delta E_{2000}$ colour difference, forward renderings based on the recovered properties were indistinguishable from those preset, which were obtained from integrating sphere measurements on real materials. The recovered optical properties showed less than 4% difference relative to these measurements. This work demonstrates a versatile approach for optical material characterisation, with significant potential for digital twin creation and soft-proofing in manufacturing. Full article

In nature, some tomato (Solanum lycopersicum) shapes appear to be ellipsoidal. This study aims to fit the ellipsoid tomato profile using explicit Preston’s equation (EPE), and calculate its volume (V_pred) and surface area (S) based on the estimated EPE’s parameters. This method offers low-cost and non-destructive advantages compared to three-dimensional (3D) scanning. A total of 917 tomatoes from three cultivars were photographed, and the two-dimensional (2D) boundary coordinates of each fruit profile were digitized and then fitted using EPE. The results demonstrated that the EPE effectively fitted the tomato 2D-profile, with truss tomato ranking highest, followed by cherry, and then Qianxi. A significant relationship was found between V_pred and observed volume (V_obs) at the cultivar level. The 95% confidence intervals for the slopes for cherry tomatoes include 1.0, and for Qianxi were close to 1.0, which confirmed that these two cultivars were solids of revolution. Additionally, for cherry and Qianxi tomato, S is proportional to the V_obs (i.e., S∝V_obs^0.62~0.63), V_pred is proportional to (LW²)^0.73~0.74, and S is proportional to (LW²)^0.49 (L is the length and W is the maximum width). For any isometrically scaling solid of revolution, the theoretical exponent of surface area to volume is exactly 2/3. The observed exponent of 0.62–0.63 is a biological reality, which reveals that evolution has shaped organisms not for geometric similarity, but for functional optimization. This study can be extended to a geometry study on other egg-shaped fruits and provides a potentially simple method for calculating volume and surface area based on photographed 2D fruit profiles. Full article

We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction–diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction–diffusion master equation (RDME) representation in regions of low concentration and continuum partial differential equations (PDEs) where concentrations are high, dynamically switching based on local thresholds. This is an advancement over the existing methods in the literature, requiring no fixed spatial interfaces, enabling efficient and accurate simulation of systems in which stochasticity plays a key role but is not required uniformly across the domain. We specify the full mathematical formulation of the SRCM, including conversion reactions, hybrid kinetic rules, and consistent numerical updates. The method is validated across several one-dimensional test systems, including simple diffusion from a region of high concentration, the formation of a morphogen gradient, and the propagation of FKPP travelling waves. The results show that the SRCM captures key stochastic features while offering substantial gains in computational efficiency over fully stochastic models. Full article

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant. Full article

A truncated M-fractional Kadomtsev–Petviashvili (KP) equation in high-dimensional space has been proposed. The model provides the oretical support for studying the interaction patterns among waves. According to the attributes of the truncated M-fractional derivative, the truncated M-fractional KP equation can be reduced to the new extended KP equation. New interaction patterns which are compositions of cnoidal functions and soliton or trigonometric functions have been derived by the consistent Riccati expansion method. Applying the simple direct method, the finite symmetry transformation group and Bäcklund transformation have been constructed. Based on the known dark soliton solution and lump solution, new interaction patterns have been derived, including compositions of a dark soliton and an exponential function, compositions of a dark soliton and a trigonometric sine function, and compositions of a lump and a trigonometric sine function. The innovative aspect lies in the way that we find two effective ways to construct new interplay patterns of fractional differential equations. Full article

This work investigates the solutions of fractional integro-differential equations (FIDEs) using a unique kernel operator within the Caputo framework. The problem is addressed using both analytical and numerical techniques. First, the two-step Adomian decomposition method (TSADM) is applied to obtain an exact solution (if it exists). In the second part, numerical methods are used to generate approximate solutions, complementing the analytical approach based on the Adomian decomposition method (ADM), which is further extended using the Sumudu and Shehu transform techniques in cases where TSADM fails to yield an exact solution. Additionally, we establish the existence and uniqueness of the solution via fixed-point theorems. Furthermore, the Ulam–Hyers stability of the solution is analyzed. A detailed error analysis is performed to assess the precision and performance of the developed approaches. The results are demonstrated through validated examples, supported by comparative graphs and detailed error norm tables ($L_{\infty }$, $L_2$, and $L_1$). The graphical and tabular comparisons indicate that the Sumudu-Adomian decomposition method (Sumudu-ADM) and the Shehu-Adomian decomposition method (Shehu-ADM) approaches provide highly accurate approximations, with Shehu-ADM often delivering enhanced performance due to its weighted formulation. The suggested approach is simple and effective, often producing accurate estimates in a few iterations. Compared to conventional numerical and analytical techniques, the presented methods are computationally less intensive and more adaptable to a broad class of fractional-order differential equations encountered in scientific applications. The adopted methods offer high accuracy, low computational cost, and strong adaptability, with potential for extension to variable-order fractional models. They are suitable for a wide range of complex systems exhibiting evolving memory behavior. Full article

In this article, we present the extended simple equations method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The expansions made to the original SEsM algorithm are implemented in several directions: (1) In constructing analytical solutions: exact solutions to FNPDE systems are presented by simple or complex composite functions, including combinations of solutions to two or more different simple equations with distinct independent variables (corresponding to different wave velocities); (2) in selecting appropriate fractional derivatives and appropriate wave transformations: the choice of the type of fractional derivatives for each system of FNPDEs depends on the physical nature of the modeled real process. Based on this choice, the range of applicable wave transformations that are used to reduce FNPDEs to nonlinear ODEs has been expanded. It includes not only various forms of fractional traveling wave transformations but also standard traveling wave transformations. Based on these methodological enhancements, a generalized SEsM algorithm has been developed to derive exact solutions of systems of FNPDEs. This algorithm provides multiple options at each step, enabling the user to select the most appropriate variant depending on the expected wave dynamics in the modeled physical context. Two specific variants of the generalized SEsM algorithm have been applied to obtain exact solutions to two time-fractional shallow-water-like systems. For generating these exact solutions, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. As a result, numerous novel multi-wave solutions are derived, involving combinations of hyperbolic-like, elliptic-like, and trigonometric-like functions. The obtained analytical solutions can provide valuable qualitative insights into complex wave dynamics in generalized spatio-temporal dynamical systems, with relevance to areas such as ocean current modeling, multiphase fluid dynamics and geophysical fluid modeling. Full article

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations. Full article

The matrix sign function has a key role in several applications in numerical linear algebra. This paper presents a novel iterative approach with a sixth order of convergence to efficiently compute this function. The scheme is constructed via the employment of a nonlinear equations solver for simple roots. Then, the convergence of the extended matrix procedure is investigated to demonstrate the sixth rate of convergence. Basins of attractions for the proposed solver are given to show its global convergence behavior as well. Finally, the numerical experiments demonstrate the effectiveness of our approach compared to classical methods. Full article

In this paper, the local and global structure of positive solutions for a general predator–prey model in a multi-dimension with ratio-dependent predator influence and prey taxis is investigated. By analyzing the corresponding characteristic equation, we first obtain the local stability conditions of the positive equilibrium caused by prey taxis. Secondly, taking the prey-taxis coefficient as a bifurcation parameter, we obtain the local structure of the positive solution by resorting to an abstract bifurcation theorem, and then extend the local solution branch to a global one. Finally, the local stability of such bifurcating positive solutions is discussed by the method of the perturbation of simple eigenvalues and spectrum theory. The results indicate that attractive prey taxis can stabilize positive equilibrium and inhibits the emergence of spatial patterns, while repulsive prey taxis can lead to Turing instability and induces the emergence of spatial patterns. Full article

In this study, we propose a generalized framework based on the Simple Equations Method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The key developments over the original SEsM in the proposed analytical framework include the following: (1) an extension of the original SEsM by constructing the solutions of the studied FNPDEs as complex composite functions which combine two single composite functions, comprising the power series of the solutions of two simple equations or two special functions with different independent variables (different wave coordinates); (2) an extension of the scope of fractional wave transformations used to reduce the studied FNPDEs to different types of ODEs, depending on the physical nature of the studied FNPDEs and the type of selected simple equations. One variant of the proposed generalized SEsM is applied to a mathematical generalization inspired by the classical Boussinesq model. The studied time-fractional Boussinesq-like system describes more intricate or multiphase environments, where classical assumptions (such as constant wave speed and energy conservation) are no longer applicable. Based on the applied SEsM variant, we assume that each system variable in the studied model supports multi-wave dynamics, which involves combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous new multi-wave solutions including combinations of different hyperbolic, elliptic, and trigonometric functions are derived. To visualize the wave dynamics and validate the theoretical results, some of the obtained analytical solutions are numerically simulated. The new analytical solutions obtained in this study can contribute to the prediction and control of more specific physical processes, including diffusion in porous media, nanofluid dynamics, ocean current modeling, multiphase fluid dynamics, as well as several geophysical phenomena. Full article

The reliance of feedback mechanisms in conventional light-fueled self-oscillating systems on spatially distributed light and intricately designed structures impedes their application and development in micro-robots, miniature actuators, and other small-scale devices. This paper presents a straightforward rheostat feedback mechanism to create an electrically driven liquid crystal elastomer (LCE) self-oscillator which comprises an LCE fiber, a rheostat, a spring, and a mass. Based on the electrothermally responsive LCE model, we first derive the governing equation for the system’s dynamics and subsequently formulate the asymptotic equation. Numerical calculations reveal two motion phases, i.e., static and self-oscillating, and elucidate the mechanism behind self-oscillation. By employing the multi-scale method, we identify the Hopf bifurcation and establish the analytical solutions for amplitude and frequency. The influence of various system parameters on the amplitude and frequency of self-oscillation was analyzed, with numerical solutions being validated against analytical results to ensure consistency. The proposed rheostat feedback mechanism can be extended to cases with rheostats that have more general resistance properties and offers advantages such as simple design, adjustable dimensions, and rapid operation. The findings are expected to inspire broader design concepts for applications in soft robotics, sensors, and adaptive structures. Full article

In this paper, I extend the Simple Equations Method (SEsM) and adapt it to obtain exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). The novelty in the extended SEsM algorithm is that, in addition to introducing more simple equations in the construction of the solutions of the studied FNPDEs, it is assumed that the selected simple equations have different independent variables (i.e., different coordinates moving with the wave). As a consequence, nonlinear waves propagating with different wave velocities will be observed. Several scenarios of the extended SEsM are applied to the time-fractional predator–prey model under the Allee effect. Based on this, new analytical solutions are derived. Numerical simulations of some of these solutions are presented, adequately capturing the expected diverse wave dynamics of predator–prey interactions. Full article

This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least $7.993$, attaining the efficiency index $7.{993}^{1/3}\approx 2$. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form $z^n-1$ in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods. Full article

The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivatives do not exist. However, the method can converge. This indicates that the most sufficient convergence conditions required by the Taylor approach can be replaced by weaker ones. Other drawbacks exist, such as information on the isolation of simple solutions or the number of iterations that must be performed to achieve the desired error tolerance. This paper positively addresses all these issues by considering a technique that uses only the operators on the method and Ω-generalized continuity to control the derivative. Moreover, both local and semi-local convergence analyses are presented for Banach space-valued operators. The technique can be used to extend the applicability of other methods along the same lines. A large number of concrete examples are shown in which the convergence conditions are fulfilled. Full article