Homotopy Parameter Research Articles

Novel approaches for nonlinear Sine-Gordon equations using two efficient techniques

ABSTRACT In this work, we obtained a new functional matrix using Clique-polynomials of complete graphs K n with n vertices and considered a new approach to solving the Sine–Gordon (SG) equation. The clique polynomial method transforms this equation into a system of algebraic equations. The solution will be drawn with the help of Newton Raphson’s method. Also, we employed the q-homotopy analysis transform method (q-HATM), which is the proper collision of the Laplace transform and the q-homotopy analysis method (q-HAM). To witness the reliability and accuracy of the considered schemes, some illustrations of the SG equation and double SG equation are considered. Here, the SG equation is solved easily and elegantly without using discretization or transformation of the equation by using the q-HATM. Also, in q-HATM, the presence of homotopy and axillary parameters allows us to have a large convergence region. The 3D surfaces of acquired solutions are drawn effectively. The tables of error analysis demonstrate the success of these methods.

Solving Mathematical Programs with Complementarity Constraints Arising in Nonsmooth Optimal Control

AbstractThis paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call . The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes’ relaxation (SIAM J. Optim. 11, 918–936, 2001) with (Math. Program. 106, 25–57, 2006) as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.

Fractional Approach for Belousov-Zhabotinsky Reactions Model with Unified Technique

The Belousov-Zhabotinsky reaction model represents chemical oscillators that exhibit periodic vibrations as a result of complex physic-chemical phenomena. The non-linear behaviour exhibited by Belousov-Zhabotinsky model is the cause of Turing patterns, birth of spiral waves, rise of limit cycle attractors, and deterministic chaos in many chemical reaction processes. Due to these noteworthy characteristics, in this paper, we have analyzed mathematical Belousov-Zhabotinsky model by a novel numerical approach q-Homotopy analysis transformation method. To interpret new observations, we have incorporated Caputo fractional derivative in the model. The numerical result are presented graphically and concerning the absolute error of solutions. With the help of the homotopy parameter curve, we have projected the convergence region with reference to diverse values of fractional derivative. This work establishes that the projected numerical algorithm is a well-organized tool to analyze the multifaceted coupled partial differential equation representing Belousov-Zhabotinsky type reactions.

Upper and lower solutions method for a class of second-order coupled systems

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.

A Generalized Hybrid Method for Handling Fractional Caputo Partial Differential Equations via Homotopy Perturbed Analysis

This article describes a novel hybrid technique known as the Sawi transform homotopy perturbation method for solving Caputo fractional partial differential equations. Combining the Sawi transform and the homotopy perturbation method, this innovative technique approximates series solutions for fractional partial differential equations. The Sawi transform is a recently developed integral transform that may successfully manage recurrence relations and integro-differential equations. Using a homotopy parameter, the homotopy perturbation method is a potent semi-analytical tool for constructing approximate solutions to nonlinear problems. The suggested method offers various advantages over existing methods, including high precision, rapid convergence, minimal computing expense, and broad applicability. The new method is used to solve the convection–reaction–diffusion problem using fractional Caputo derivatives.

Differential contracting homotopy in higher-spin theory

A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables ZA and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.

Accelerating Deep Neural Network training for autonomous landing guidance via homotopy

The feasibility of utilizing Deep Neural Networks for generating guidance commands in spacecraft landing scenarios has garnered recent research interest. A primary challenge lies in the inefficiency of training sample generation. This paper proposes a new approach to address it by utilizing nonlinear homotopy methods, which involve embedding the homotopic parameter either into the performance index or the right-hand side of the differential equations. Unlike the existing approaches that overlooked the potential value of the intermediate solutions during homotopy process, the proposed method incorporates these discarded solutions as supplementary training data, which treats the homotopic parameter and state as inputs, and the control as the output. Two examples, namely a minimum-fuel lunar landing and a minimum-time asteroid landing, are provided to illustrate the effectiveness and advantages of the proposed method.

Bayesian ensemble machine learning-assisted deterministic and stochastic groundwater DNAPL source inversion with a homotopy-based progressive search mechanism

A hyperheuristic homotopy algorithm (HH-HA) and a homotopy-based swarm intelligence algorithm for parallel stochastic search are proposed to improve the search ergodicity and stability of deterministic and stochastic inversion approaches, respectively, for the identification of dense nonaqueous phase liquid (DNAPL) source characteristics and contaminant transport parameters in groundwater. Meanwhile, a Bayesian ensemble machine learning (BEML) method for numerical simulation surrogate modeling is proposed to enhance the learning and generalization capability, while significantly reducing the computational cost of repetitive CPU-demanding likelihood evaluations in inversion iterations. Two cases are presented for the verification and application of the proposed inverse modeling methods. The results show that the BEML surrogate model is powerful in approximating the numerical model outputs, reducing the mean relative errors of the contaminant concentration and pressure head predictions to 1.18% and 1.29%, respectively. The homotopy-based progressive search mechanism is highly effective in achieving more accurate identification values for deterministic inversion and a more reliable posterior distribution for stochastic inversion. The HH-HA shows stable performance, approaching the global optimum in wide areas and reducing the mean relative error of identification from 11.95% to 4.11%. The homotopy-based parallel stochastic search lines adequately cover the search space tracking the change of homotopy parameter, and the mean relative error of point estimation is reduced to 2.03%. Though the deterministic inversion approach can provide intuitive identification values, the reliability of the inversion outputs cannot be quantified. In contrast, the proposed stochastic inversion system is of great significance for providing decision makers with comprehensive reference information.

Fast Homotopy for Spacecraft Rendezvous Trajectory Optimization with Discrete Logic

This paper presents a computationally efficient optimization algorithm for solving nonconvex optimal control problems that involve discrete logic constraints. Traditional solution methods require binary variables and mixed-integer programming (MIP), which is prohibitively slow and computationally expensive. This paper proposes a faster and computationally cheaper algorithm that can produce locally optimal solutions in seconds. This is achieved by blending sequential convex programming and numerical continuation into a single iterative solution process. The algorithm approximates discrete logic constraints with smooth functions and uses a homotopy parameter to control the accuracy of this approximation. The homotopy parameter is updated such that, by the time the algorithm converges, the smooth approximations enforce the exact discrete logic. The effectiveness of this approach is numerically demonstrated for a realistic rendezvous scenario inspired by the Apollo Transposition and Docking maneuver. In less than 15 s of cumulative solver time, the algorithm finds a fuel-minimizing trajectory that obeys the following discrete logic constraints: thruster minimum impulse-bit, range-triggered approach cone, and range- triggered plume impingement. The optimized trajectory uses significantly less fuel than reported NASA design targets.

Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm.

On dynamical behavior for approximate solutions sustained by nonlinear fractional damped Burger and Sharma–Tasso–Olver equation

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

The exploration of thin liquid film flow under the influences of various physical parameters by a new analytical innovation

AbstractA new analytical innovation is utilized to explore the thin liquid film flow of magnetohydrodynamic (MHD) fluid over an unsteady porous stretching surface. Here magnetic field, thermal radiation, and variable viscosity are taken. The self‐similarity variables have been used to transform the modeled partial differential equations into a set of non‐linear coupled differential equations. These non‐linear differential equations for the velocity and temperature profiles have been tackled through an innovation containing homotopy, auxiliary functions, and convergence control parameters. This method is called the second configuration of the Optimal Homotopy Asymptotic Method and is denoted by (OHAM‐2). Galerkin's process is chosen for the optimizations of parameters. The achieved coupled equations are also solved numerically (ND‐Solve Method), and the obtained outcomes are compared with the results elaborated by the suggested algorithm. Here the utmost attention is on the rapid convergence of the proposed algorithm and the consequences of various tangible variables on the velocity and temperature profiles. This straightforward algorithm consists of a few steps but gives better outcomes. The technique can be applied to solve partial differential equations and their systems forming in various disciplines. This technique can also solve Integro differential equations. The residuals achieved by the proposed method for the velocity and temperature fields are shown graphically. The errors and residuals are also taught in Table 1 and Table 2. The effects of parameters are inculcated in Table 3 and Table 4. The results are validated with the published work as shown in Table 5. Nomenclature is indicated in the text below.

A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1 + 2)-physical models

Abstract In the present article, we geometrically and analytically examine the mutual impact of space-time Caputo derivatives embedded in (1 + 2)-physical models. This has been accomplished by integrating the residual power series method (RPSM) with a new trivariate fractional power series representation that encompasses spatial and temporal Caputo derivative parameters. Theoretically, some results regarding the convergence and the error for the proposed adaptation have been established by virtue of the Riemann–Liouville fractional integral. Practically, the embedding of Schrödinger, telegraph, and Burgers’ equations into higher fractional space has been considered, and their solutions furnished by means of a rapidly convergent series that has ultimately a closed-form fractional function. The graphical analysis of the obtained solutions has shown that the solutions possess a homotopy mapping characteristic, in a topological sense, to reach the integer case solution where the Caputo derivative parameters behave similarly to the homotopy parameters. Altogether, the proposed technique exhibits a high accuracy and high rate of convergence.

Estimation of the Traveling Wave Speed in Pancreatic Islets Using High Period Periodic Orbits

In response to an increase in blood glucose levels, insulin is released into the bloodstream by the pancreatic islets of Langerhans. As a result of this influx of glucose, the islets start what are called bursting oscillations of the membrane potential and the intracellular calcium concentration. Time delays of several seconds in the activity of distant cells in the islets have been observed, indicating the presence of traveling waves through the islets. By considering a robust model of a pancreatic islet in one dimension, we study the relationship between the wave speed and the model parameters for the existence of traveling wave fronts and traveling wave pulses. After a systematic reduction of the model equations, the wave fronts (or heteroclinic connection) are studied. Using the bi-stable equation, for which an exact expression of the heteroclinic connection can be computed, we use a homotopy parameter to move from this equation to an islet model. A relationship between the wave speed and the conductance of the ATP-modulated potassium channel is constructed. Upon the inclusion of the slow gating variable back into the model equations, we observe the presence of a traveling wave pulse (or homoclinic connection). Using a high period periodic orbit to approximate the homoclinic orbit, a similar relationship between these two parameters is constructed. We observe that the heteroclinic connection is a good approximation for a portion of the homoclinic connection. Comparisons of the speed of the wave traveling through the islet in the partial differential equation model and the model in traveling coordinates is carried out.

Combining Uneliminated Algebraic Formulations With Sparse Linear Solvers to Increase the Speed and Accuracy of Homotopy Path Tracking for Kinematic Synthesis

Abstract The method of kinematic synthesis requires finding the solution set of a system of polynomials. Parameter homotopy continuation is used to solve these systems and requires repeatedly solving systems of linear equations. For kinematic synthesis, the associated linear systems become ill-conditioned, resulting in a marked decrease in the number of solutions found due to path tracking failures. This unavoidable ill-conditioning places a premium on accurate function and matrix evaluations. Traditionally, variables are eliminated to reduce the dimension of the problem. However, this greatly increases the computational cost of evaluating the resulting functions and matrices and introduces numerical instability. We propose avoiding the elimination of variables to reduce required computations, increasing the dimension of the linear systems, but resulting in matrices that are quite sparse. We then solve these systems with sparse solvers to save memory and increase speed. We found that this combination resulted in a speedup of up to 250 × over traditional methods while maintaining the same accuracy.

Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations

In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.

Forward Statics of Tensegrity Robots With Rigid Bodies Using Homotopy Continuation

Tensegrity robots are composed of simple members (rigid bodies and tensile cables) and have features like lightweight, compliance, and robustness, thus representing a promising alternative to soft and rigid robots. Forward statics, which determines the robot pose in static equilibrium, is very helpful for the design of tensegrity robots. However, existing approaches are not efficient or not general enough for Class-<inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> tensegrity robots, where up to <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> rigid bodies can be connected. This letter proposes a general approach for static modeling of tensegrity robots with rigid bodies of arbitrary shapes, using natural coordinates. Introducing an additional set of variables transforms the static equation into a polynomial system with four kinds of parameters: cable rest lengths, deformations, stiffness coefficients, and external forces. Then, the forward statics problem appears as a path-following problem of a parameter homotopy, which is efficiently solved by the homotopy continuation method. Simulations illustrate how the forward statics can be used to evaluate the achievable range and the strength of a 2D tensegrity manipulator and a 3D tensegrity spine. Finally, we conduct hardware experiments on a tensegrity manipulator prototype, validating the accuracy of the proposed approach.

A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators

The Korteweg–De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface critical flow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional operators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physical nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order.

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.