Homotopy Map Research Articles

Flow of magnetohydrodynamic blood-based hybrid nanofluids with double diffusion in the presence of Riga plate for heat optimization and drug applications

In a recent study, researchers investigated the flow behavior of Casson Hybrid nanofluids (HNFs) combination of single and multi-walled carbon nanotubes (SWCNTs), (MWCNTs) on a Riga plate for drug delivery applications. The study found that the Casson HNFs exhibited non-Newtonian behavior on the Riga plate, with the presence of nanoparticles causing an increase in viscosity and shear-thinning behavior. This rheological behavior is favorable for drug delivery applications as it improves the stability and dispersion of drug particles in the fluid. The similarity equations of the flow problem are easily tackled with the homotopy analysis method (HAM) built on fundamental homotopy mapping. In high-speed flows, Riga actuators are expected to achieve the requirements, since HNF is enhanced by modified Hartmann numbers. As the Eckert number, heat generation/absorption parameter, and thermal relaxation time parameter decrease the temperature, thermal transport increases. Furthermore, with the increments in paramount parameters, the skin friction coefficient and heat transfer rate are remarkably meliorated under higher modified Hartmann number. Furthermore, the study also found that the Casson Hybrid nanofluids showed enhanced heat transfer properties on the Riga plate, which is beneficial for localized drug delivery applications that require precise temperature control.

Hybrid nanofluids flow over a Riga plate surrounded by a variable porous medium for heat transfer optimization

Enhancing the heat transfer rate in hybrid nanofluids is significantly aided by a porous medium. The nanofluid’s heat transfer to the surrounding medium is more efficient due to the presence of a porous medium. The porous space also provides additional surface contact points for heat exchange and an intricate network of interconnected pores. This article elaborates on the incorporation of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) into water to perform hybrid nanofluids. The flow is considered over a Riga plate surrounded by a variable porous space. Various applications, including thermal management systems, microelectronics cooling, and energy conversion devices, benefit greatly from the study of hybrid nanofluid flow on a Riga plate. The similarity equations of the flow problem are easily tackled with the homotopy analysis method (HAM) built on fundamental homotopy mapping. Furthermore, with the increments in paramount parameters, the skin friction coefficient and heat transfer rate are remarkably meliorated under a higher modified Hartmann number. All reverberations are illustrated in graphs and Tables. From the obtained results it is observed that flow control capabilities are enabled by the variations in permeability in the case of hybrid nanofluids. It is also observed that hybrid nanofluids are more capable to enhance the thermal capabilities of the traditional fluids as compared to the mano nanofluids.

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.

Use DG-methods to build a matrix factorization

Let P be a commutative Noetherian ring, 𝔠 be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and P¯ be the hypersurface ring P∕(f). Assume that 𝔠:f is a grade four Gorenstein ideal of P. We give a resolution N of P¯∕𝔠P¯ by free P¯-modules. The resolution N is built from a differential graded algebra resolution of P∕(𝔠:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.

The resolution of (xN,yN,zN,wN)

Let k be a field of characteristic zero, n<N be positive integers, P be the polynomial ring k[x,y,z,w], F be the homogeneous polynomial xn+yn+zn+wn, K be the ideal (xN,yN,zN,wN), and P‾ be the hypersurface ring P‾=P/(F). We describe the minimal multi-homogeneous resolution of P‾/KP‾ by free P‾-modules, the socle degrees of P‾/KP‾, and the minimal multi-homogeneous resolution of the Gorenstein ring P/(K:F) by free P-modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P‾/KP‾ by free P‾-modules is obtained from a Differential Graded Algebra resolution of P/(K:F) by free P-modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal.

A HYDRODYNAMICAL HOMOTOPY CO-MOMENTUM MAP AND A MULTISYMPLECTIC INTERPRETATION OF HIGHER-ORDER LINKING NUMBERS

AbstractIn this paper a homotopy co-momentum map (à la Callies, Frégier, Rogers and Zambon) transgressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden, Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids, and in particular of Brylinski’s manifold of smooth oriented knots, is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher-order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot-theoretic analogues of first integrals in involution are determined.

Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics

Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger’s equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

An aggregate homotopy method for solving unconstrained minimax problems

ABSTRACT In this paper, an aggregate homotopy method is proposed to solve unconstrained minimax problems. The homotopy mapping is constructed by the linear homotopy and the aggregate function for the objective function, and the homotopy parameter is also used as the smoothing parameter of the aggregate function. Under some general assumptions, the existence and global convergence of a smooth homotopy path are proved for almost all starting points in or a ball set, and a stationary point of the unconstrained minimax problem can be obtained. A path-following procedure is introduced to numerically trace the homotopy path. When the smoothing parameter becomes small enough, an alternative strategy of the path-following procedure is given to reduce the ill-conditioning of the aggregate function, it uses the Newton method to solve a nonlinear system, which is derived from the first order optimality conditions for the unconstrained minimax problem. Numerical results show that the proposed method is efficient and robust.

A differentiable path-following algorithm for computing perfect stationary points

This paper is concerned with the computation of perfect stationary point, which is a strict refinement of stationary point. A differentiable homotopy method is developed for finding perfect stationary points of continuous functions on convex polytopes. We constitute an artificial problem by introducing a continuously differentiable function of an extra variable. With the optimality conditions of this problem and a fixed point argument, a differentiable homotopy mapping is constructed. As the extra variable becomes close to zero, the homotopy path naturally provides a sequence of closely approximate stationary points on perturbed polytopes, and converges to a perfect stationary point on the original polytope. Numerical experiments are implemented to further illustrate the effectiveness of our method.

Infinitesimal Chow dilogarithm

Let C 2 C_{2} be a smooth and projective curve over the ring of dual numbers of a field k . k. Given non-zero rational functions f , g , f,g, and h h on C 2 , C_{2}, we define an invariant ρ ( f ∧ g ∧ h ) ∈ k . \rho (f\wedge g \wedge h) \in k. This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm of [Algebra Number Theory 3 (2009), pp. 1–34]. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture of Goncharov [J. Amer. Math. Soc. 18 (2005), pp. 1–60] with an explicit formula for the homotopy map. Also using ρ , \rho , we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park’s construction in the case of cycles with modulus [Amer. J. Math. 131 (2009), pp. 257–276].

A differentiable homotopy method to compute perfect equilibria

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

A Reformulation-Based Simplicial Homotopy Method for Approximating Perfect Equilibria

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975). A well-known application of this concept is that every perfect equilibrium of the agent normal form game of an extensive form game with perfect recall yields a trembling-hand perfect equilibrium (consequently a sequential equilibrium). To compute a perfect equilibrium, this paper extends Kohlberg and Mertens’s equivalent reformulation of Nash equilibrium to a perturbed game. This extension naturally leads to a homotopy mapping on the Euclidean space. With this homotopy mapping and a triangulation, we develop a simplicial homotopy method for approximating perfect equilibria. It is proved that every limit point of the simplicial path yields a perfect equilibrium. Numerical results further confirm the effectiveness of the method.

A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities

This paper presents a smooth path-following algorithm for computing market equilibrium in a pure exchange economy under a class of piecewise-smooth concave utilities, which can be expressed as $$u(x)=\min _\ell \{f_\ell (x)\}$$ with $$f_\ell (x)$$ being a smooth concave function for all $$\ell $$ . As a result of a smooth technique for minimax problems, a smooth homotopy mapping is derived from the introduction of logarithmic barrier terms and an extra variable. With this mapping, it is proved that there always exists a smooth path leading to a market equilibrium as the extra variable approaches zero. A predictor–corrector method is adapted for numerically following this path. Numerical results are given to further demonstrate the effectiveness and efficiency of the algorithm.

Simulation of incompressible two-phase flow in porous media with large timesteps

Multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering, petroleum systems' analysis, and CO2 sequestration. While simulations often use a fully implicit discretization to increase the time step size, restrictions on the time step often exist due to non-convergence of the nonlinear solver (e.g. Newton's method). Here this problem is addressed for the Buckley–Leverett equations, which model incompressible, immiscible, two-phase flow with no capillary potential. The equations are recast as a gradient flow using the phase-field method, and a convex energy splitting scheme is applied to enable large timesteps, even for high degrees of heterogeneity in permeability and viscosity. By using the phase-field formulation as a homotopy map, the underlying hyperbolic flow equations can be solved with large timesteps. For a heterogeneous test problem, the new homotopy method allows the timestep to be increased by more than six orders of magnitude relative to the unmodified equations while maintaining convergence.

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.

A complex line homotopy algorithm for coupled algebraic Riccati equations

This paper presents an algorithm based on a complex line homotopy mapping for the numerical solution of systems of coupled algebraic matrix Riccati equations which often met in continuous time control system design problems. Well-defined complex homotopy paths lead to solutions globally from independent initial conditions, and they are computed via continuous dynamic flow algorithms (ordinary differential equation solvers). Although the homotopy mapping is complex it is shown how it is possible in practice to obtain desired real solutions of certain definiteness even from zero initial conditions. Moreover, a scaling method is deployed within the algorithms for numerical enhancement and its effect to homotopy paths, convergence and stability is experimentally studied. In addition, a first-order perturbation analysis framework for the assessment of computed solutions is provided. Comparisons with other existing methods from the literature are made through several numerical examples considering mixed H|$_{2}$|/H|$_{\infty }$|⁠, Nash games and stochastic control system design problems, where the efficiency of the proposed algorithm is illustrated. It is worth noticing that algorithmic implementation and numerical experimentation with complex homotopies for the coupled algebraic Riccati equations under study appears to be absent the literature. Hence, the contribution of the present paper to related problems in systems and control.

The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

Robustness of declarative modeling languages: Improvements via probability-one homotopy

Robustness issues with steady-state initialization remain a barrier in the practical use of declarative modeling languages for multi-domain modeling of large, complex, and heterogeneous technical systems. The objective of this paper is to illustrate how probability-one homotopy, an established method from topology, can solve this issue. This is achieved by establishing a framework for application-specific probability-one homotopy in declarative modeling languages. The analysis is based on domain-specific probability-one homotopy maps, which were reformulated in a declarative fashion. Additionally, a novel probability-one homotopy map and associated coercivity proof is introduced for a class of thermo-fluid dynamics problems. It was found that the approach enables robust initialization for declarative modeling languages on several test cases and leads to a concise declarative problem formulation.

Multiple solutions for fractional differential equations: Analytic approach

Multiple solutions of the fractional differential equations is an interesting subject in the area of mathematics, sciences and engineering. A new Algorithm for finding multiple solution of fractional differential equations is constructed based on a homotopy map between initial approximation and exact solution with predictor force condition. Easy and efficient algorithm is introduced to approximate the multiple solutions, even if these multiple solutions are very close and thus rather difficult to distinct even by numerical techniques. Several examples are presented to demonstrate the efficiency of the algorithm. To the best of our knowledge, we present multiple solutions for fractional differential equations analytically.

A Globally Convergent Probability-One Homotopy for Linear Programs with Linear Complementarity Constraints

A solution of the standard formulation of a linear program with linear complementarity constraints (LPCC) does not satisfy a constraint qualification. A family of relaxations of an LPCC, associated with a probability-one homotopy map, proposed here is shown to have several desirable properties. The homotopy map is nonlinear, replacing all the constraints with nonlinear relaxations of NCP functions. Under mild existence and rank assumptions, (1) the LPCC relaxations RLPCC($\lambda$) have a solution for $0\le\lambda \le1$; (2) RLPCC(1) is equivalent to LPCC; (3) the Kuhn--Tucker constraint qualification is satisfied at every local or global solution of RLPCC($\lambda$) for almost all $ 0\le\lambda <1$; (4) a point is a local solution of RLPCC(1) (and LPCC) if and only if it is a Kuhn--Tucker point for RLPCC(1); and (5) a homotopy algorithm can find a Kuhn--Tucker point for RLPCC(1). Since the homotopy map is a globally convergent probability-one homotopy, robust and efficient numerical algorithms exist to f...