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This paper advances a new application of q-homotopy analysis method (q-HAM) to solve non-linear optimal control problems(NOCPs). First, the NOCP was transformed into a non-linear two-point boundary value problem by using the Pontryagins maximum principle (PMP). Then, we applied the q-HAM to solve this system. The proposed method is based on the HAM but the q-HAM, has an increased interval of convergence than the HAM. Three examples are provided to demonstrate the reliability and efficiency of the method. Next, the numerical results of the proposed method are compared with those of other methods. As can be seen from the tables, the maximum error in the second and third examples is much better than other methods. [ABSTRACT FROM AUTHOR]

The Padé approximation and its enhancements provide a more accurate approximation of functions than the Taylor series truncation. A new technique for approximating functions into rational functions is proposed in this paper. This technique is based on the homotopy Padé technique and introduces new parameters known as merging parameters. These parameters are added to the Tayler series before the Padé process is computed. To control error, the merging parameters and dividing the interval into subintervals are used. Two illustrative examples are used to demonstrate the validity and reliability of the proposed novel approximation. The robustness and efficiency of the proposed approximation were demonstrated by computing the absolute error and comparing the results to those of the standard Padé technique and the generalized restrictive Padé technique. Also, Hard-core scattering problem and Debye-Hukel function are tested by the proposed technique. The piecewise homotopy Padé method is an excellent path to approximate any function. The proposed new approximation's efficacy and accuracy have been validated using Mathematica 12. [ABSTRACT FROM AUTHOR]

In this paper, a time-fractional Cauchy equation (TFCE) is analyzed by using the q-homotopy analysis Shehu transform algorithm (q-HASTA) with convergence analysis. The q-HASTA comprises with the reduced differential transform algorithm (RDTA). The solution of TFCE is represented in the series form by using the q-HASTA scheme. The TFCE is transformed into algebraic form for finding the general solution efficiently. This provides a compact form solution with minimized error. There are three key outcomes of the work. First, the small size of input parameters by the RDTA transforms into the subsidiary equation so that it takes short time to solve. As the second advantage, the structure of the problem is reduced by controlling the solution series; hence the characterization of the solution becomes classified for finding the particular solution. The third advantage of this work is that the approximate solution with absolute error approximation for the fractional model of the problem is handled by using a generalized and efficient scheme q-HASTA. These outcomes are illustrated by graphs and tables. [ABSTRACT FROM AUTHOR]

With effective techniques like the homotopy perturbation approach and the Adomian decomposition method via the Yang transform, the time-fractional vibration equation's solution is found for large membranes. In Caputo's sense, the fractional derivative is taken. Numerical experiments with various initial conditions are carried out through a few test examples. The findings are described using various wave velocity values. The outcomes demonstrate the competence and reliability of this analytical framework. Figures are used to discuss the solution of the fractional vibration equation using the suggested strategies for different orders of memory-dependent derivative. The suggested approaches reduce computation size and time even when the accurate solution of a nonlinear differential equation is unknown. It is helpful for both small and large parameters. The results show that the suggested techniques are trustworthy, accurate, appealing and effective strategies. [ABSTRACT FROM AUTHOR]

The major goal of this research is to use a new integral transform approach to obtain the exact solution to the time-fractional convection-reaction-diffusion equations (CRDEs). The proposed method is a combination of the Elzaki transform and the homotopy perturbation method. He's polynomial is used to tackle the nonlinearity which arise in our considered problems. Three test examples are considered to show the accuracy of the proposed scheme. In order to find satisfactory approximations to the offered problems, this work takes into account a sophisticated methodology and fractional operators in this context. In order to achieve better approximations after a limited number of iterations, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for CRDEs. It has been found that the proposed method's solution converges at the desired rate towards the accurate solution. We give some graphical representations of the accurate and analytical results, which are in excellent agreement with one another, to demonstrate the validity of the suggested methodology. For validity of the present technique, the convergence of the fractional solutions towards integer order solution is investigated. The proposed method is found to be very efficient, simple, and suitable to other nonlinear problem raised in science and engineering. [ABSTRACT FROM AUTHOR]

This communication describes the Homotopy Perturbation Method with Laplace Transform (LTHPM), which is used to solve the chemical kinetics problem analytically investigated. This technique is used to solve the chemical kinetics problem in bromine. Coupled nonlinear ordinary differential equations form this problem. In this perturbation method, the solution is obtained in the approximate form with easily computed components. The Laplace transform used to accelerate the convergence of the power series, and the results are shown in the graphs, which have good agreement with the other existing method in the literature. The results show that LTHPM is very effective and easy to implement. [ABSTRACT FROM AUTHOR]

The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically. [ABSTRACT FROM AUTHOR]

This paper attempts to carry out a study on a specific type of time-fractional differential equation called Benjamin-Bona-Mahony- Burger (BBM-Burger). This equation describes the mathematical model of unidirectional transmission of low-amplitude long waves through frequencydependent dispersive media. To go ahead the research, optimal homotopy asymptotic method (OHAM) and its new version (NOHAM) are applied to find analytic approximations to the BBM-Burger equation using the symbolic Maple package. Moreover, convergence of the both methods are addressed. The approximate results of the proposed methods are compared with the exact solution of BBM-Burger equation. What we learn from the results of applying OHAM and NOHAM on the BBM-Burger equation, is the high satisfactory accuracy. However, NOHAM produces an approximate solution with lower computational cost, than OHAM. [ABSTRACT FROM AUTHOR]

The proposed new technique in this article is based on modification Homotopy Perturbation method (HPM) using Hermite polynomials for approximate solution to fuzzy integral equations of the second kind for both Volterra and Fredholm types. Our method produces accurate results which are illustrated throughout some examples. [ABSTRACT FROM AUTHOR]

Adapting a result of Félix-Halperin-Lemaire concerning the Lusternik-Schnirelmann category of products, we prove the additivity of a rational approximation for Schwarz’s sectional category with respect to products of certain fibrations. [ABSTRACT FROM AUTHOR]

Abstract The q -difference equations are kind of important problems in q -calculus and applied mathematics. In this paper, the homotopy analysis method is extended to find approximate solution for some of q -differential equations. The q -diffusion equation and some examples are analytically investigated. The series solutions obtained by the proposed method are checked by reducing the solutions of q -calculus problems to h -calculus approximate solutions when q → 1. [ABSTRACT FROM AUTHOR]

In this paper, a numerical procedure called multistage optimal homotopy asymptotic method (MOHAM) is introduced to solve multipantograph equations with time delay. It was shown that the MOHAM algorithm rapidly provides accurate convergent approximate solutions of the exact solution using only one term. A comparative study between the proposed method, the homotopy perturbation method (HPM) and the Taylor matrix method are presented. The obtained results revealed that the method is of higher accuracy, effective and easy to use. [ABSTRACT FROM AUTHOR]

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge-Kutta method, it is found that the approximate solution agrees well with the numerical solution. [ABSTRACT FROM AUTHOR]

In this study, the effect of pressure gradient have been included for heat transfer in the boundary layer flow of Maxwell fluid over a flat plate.The solution of the problem is obtained with an application of algorithms of Adams Method (AM) and Gear Method (GM) with Homotopy Perturbation Method (HPM) as an approximation technique. This technique shows the outcomes of pressure gradient (m), Deborah number (α) and Prandtl number (Pr) in the boundary layer flow on temperature and velocity profiles, also the momentum and thermal boundary layer thickness and discussed. To obtain this objective, the momentum and energy equations of Maxwell are solved. The outcomes of HPM in the absence of relaxation time (λ) or Deborah number (α) and pressure gradient (m) (i.e. λ = α = m = 0) at Prandtl number Pr = 1 are in closed relation with the numerical results having the value of n∞1 is around 5. Also it is found that the system is convergent, as a whole momentum and thermal boundary layer thicknesses becomes thinner and thinner. Importantly, some cooling effects of the Maxwell fluid over a flat plate for energy profile have been observed. [ABSTRACT FROM AUTHOR]

In this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy analysis method (HAM) is employed to solve a class of variational problems (VPs). By using the so-called ħ-curves, we determine the convergence parameter ħ, which plays key role to control convergence of solution series. Also we use Pade' approximant to improve accuracy of the method. Two test example are given to clarify the applicability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose piecewise homotopy analysis method (PHAM) to extend the convergence region of the homotopy analysis solution for well-posed initial value problems. A convergence theorem of the homotopy analysis solution in the expression of polynomials in the form of Cauchy-Kowalevskaya theorem is given for the nonlinear equation which is analytical near the initial point. We also discuss the influence of the convergence-control parameter h. Our convergence result supports Liao's conjecture (1): the convergence region increases with the increasing of h(<0). PHAM solution with the symbol of arbitrary initial values has a general form on every convergent subinterval, since the formal series solution is determined by governing equation and initial values. So, we only should substitute the initial values to general form of PHAM solution to obtain the analytical expression on a target subintervals. We also present three typical examples to illustrate the effectiveness and validity of PHAM and convergence results. [ABSTRACT FROM AUTHOR]

In this article, using the homotopy renormalization method, the asymptotic analysis to a nonlinear problem on domain boundaries in convection patterns are given. In particular, by taking a variable coefficient homotopy equation, the global asymptotic solutions satisfying boundary conditions are obtained. These results are better than the existing analytic approximation solutions. [ABSTRACT FROM AUTHOR]

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we presented a reliable algorithm to solve the singularity initial value problems of the time-dependent fractional Emden-Fowler type equations by homotopy analysis method. The approximate solutions of the problems are obtained. [ABSTRACT FROM AUTHOR]

The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von Kármán [1]. This problem is reconsidered in this paper using an analytic approximation method, namely, the homotopy analysis method (HAM). Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new method to compute approximate analytical solutions for nonlinear differential equations, called the Least Squares Homotopy Perturbation Method. The method is based on the well-known Homotopy Perturbation Method and its main feature is an accelerated convergence compared to the regular Homotopy Perturbation Method. The comparison with previous results emphasizes the high accuracy of the method. [ABSTRACT FROM AUTHOR]

We develop a theory of Goodwillie calculus for functors between G -equivariant homotopy theories, where G is a finite group. We construct J -excisive approximations for any finite G -set J . These combine into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. We prove convergence results for the tree of a functor on pointed G -spaces that commutes with fixed-points, and we reinterpret the Tom Dieck-splitting as an instance of a more general splitting phenomenon that occurs for the fixed-points of the equivariant derivatives of these functors. As our main example we describe the layers of the tree of the identity functor in terms of the equivariant Spanier–Whitehead duals of the partition complexes. [ABSTRACT FROM AUTHOR]

This paper presents a semigroup approach for inverse source problems for the abstract heat equation, when the measured output data is given in subject to the integral overspecification over the spatial domain. The existence of a solution to the inverse source problem is shown in appropriate function spaces and a representation formula for the solution is proposed. Such representation permits the derivation of sufficient conditions for the uniqueness of the solution. Also an approximation method based on the optimal homotopy analysis method (OHAM) is designed, and the error estimates are discussed using graphical analysis. Moreover, we conjecture that our approach can be applied for the determination of a control parameter in an inverse problem with integral overspecialization data. The proposed algorithm is examined through various numerical examples for the reconstruction of continuous sources and the determination of a control parameter in parabolic equations. The accuracy and stability of the method are discussed and compared with several finite-difference techniques. Computational results show efficiency and high accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, Homotopy Perturbation Method (HPM) is applied to find the approximate solutions of ZK equations. It is proved that the HPM gives a powerful tool for solving a large number of nonlinear partial differential equations in mathematical physics. The solutions obtained by HPM are presented graphically. [ABSTRACT FROM AUTHOR]

In this study, we obtain the approximate soliton solution of the fractional generalized Hirota-Satsuma coupled Kortewegde Vries equation (GHS-cKdV) within the homotopy analysis method (HAM). Numerical results are successfully compared with other solutions obtained by the differential transform method (DTM) and the homotopy perturbation method (HPM). The numerical results indicate that the only few terms are sufficient to get the correct solutions. Also, the results are given by tables and figures. [ABSTRACT FROM AUTHOR]

This paper presents a modified Laplace transform homotopy perturbation method with finite boundary conditions (MLT-HPM) designed to improve the accuracy of the approximate solutions obtained by LT-HPM and other methods. To this purpose, a suitable initial approximation will be introduced, in addition, the residual error in several points of the interest interval (RECP) will be canceled. In order to prove the efficiency of the proposed method a couple of nonlinear ordinary differential equations with mixed boundary conditions, indeed, difficult to approximate, are proposed. The square residual error (S.R.E) of the proposed solutions will result to be of hundredths and tenths, requiring only a first order approximation of MLT-HPM, unlike LT-HPM, which will require more iterations for the same cases study. [ABSTRACT FROM AUTHOR]

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We present new analytical approximated solutions for the space-time fractional nonlinear partial differential coupled mKdV equation. A homotopy analysis method is considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding exact solutions of the fractional equation proposed, for the special case when the limit of the integral order of the time derivative is considered. The comparison shows a precise agreement between these solutions. [ABSTRACT FROM AUTHOR]

Given a connected CW-space X, SNT( X) denotes the set of all homotopy types [ X′] such that the Postnikov approximations X and X′ are homotopy equivalent for all n. The main purpose of this paper is to show that the set of all the same homotopy ntypes of the suspension of the wedges of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type of itself, i.e., SNT(Σ(K(ℤ, 2 a ) ∨ K(ℤ, 2 a ) ∨∙∙∙∨ K(ℤ, 2 a ))) = * for a < a < ∙∙∙ < a , as a far more general conjecture than the original one of the same n-type posed by McGibbon and Møller (in [McGibbon, C. A. and Møller, J. M., On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres, Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Lecture Notes in Math., 1509, 1992, 285-293].) [ABSTRACT FROM AUTHOR]

In this paper we propose an accelerated convergence method, which is combined with the homotopy analysis method (HAM), to solve nonlinear problems. The HAM is applied to obtain approximate expressions. According to the numbers of terms in the approximations, some ratio-control parameters are introduced in the solution expressions. By solving simultaneous algebraic equations, all artificial parameters can be optimally identified, including the so-called convergence-control parameter ℏ. Twoexamples are given to illustrate the validity of the new method. Comparison with L-P perturbation method and Runge-Kutta method reveals that the improved HAM is better than the standard HAM and applies especially to the problems with complicated nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper the optimal homotopy asymptotic method (OHAM) is employed to obtain approximate analytical solution of n th order ð n P 2 Þ linear fuzzy initial value problems (FIVPs). The convergence theorem of this method in fuzzy case is presented and proved. This method provides us with a convenient way to control the convergence of approximation series. The method is tested on n th linear FIVPs and comparisons of the exact solution that were made with numerical results showed the effectiveness and accuracy of this method. ª 2015 University of Bahrain. Publishing services by Elsevier B.V. [ABSTRACT FROM AUTHOR]

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems. [ABSTRACT FROM PUBLISHER]

In this study, we propose a hybrid method that combines the homotopy perturbation method (HPM) and Padé technique to obtain the approximate analytic solution of the Hamilton–Jacobi–Bellman equation. The truncated series solution for the HPM is suitable but only in a small domain when the exact solution is not obtained. To improve the accuracy and enlarge the convergence domain, the Padé technique is applied to the series solution for the HPM. Three examples are given to illustrate the applicability, simplicity, and efficiency of the proposed method. The results obtained are then compared with the exact solution and basic HPM. We demonstrate that this hybrid method provides an approximate analytic solution with higher accuracy than the classic HPM. [ABSTRACT FROM AUTHOR]

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine-Gordon equation (SGE). Two-dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Trajectory-based networks exhibit strong heterogeneous patterns amid human behaviors. We propose a notion of causal time-varying dynamic Bayesian network (cTVDBN) to efficiently discover such patterns. While asymmetric kernels are used to make the model better adherence to causal principles, the variations of network connectivities are addressed by an adaptive over-fitting control. Compact regularization paths are obtained by approximate homotopy to make the solution tractable. In our experiments, cTVDBN structure discovery has successfully revealed the evolution of time-varying relationships in a ring road system, and provided insights for plausible road structure improvements from a traffic flow dataset. [ABSTRACT FROM AUTHOR]

Here, we investigate the effectiveness of the optimal homotopy asymptotic method (OHAM) with a symbolic computational method for constructing the approximate solution for quantum Zakharov-Kuznetsov equation that is derived to describe the in magnetized plasma in ion acoustic waves. The results reveal that the method is explicit, effective and easy to use. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear problems. The basic ideas of this approach can be widely employed to solve other strongly nonlinear evaluation form equations arising in physics. [ABSTRACT FROM AUTHOR]

In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability. [ABSTRACT FROM AUTHOR]

This paper investigates a homotopy-based method for embedding with hundreds of thousands of data items that yields a parallel algorithm suitable for running on a distributed system. Current eigenvalue-based embedding algorithms attempt to use a sparsification of the distance matrix to approximate a low-dimensional representation when handling large-scale data sets. The main reason of taking approximation is that it is still hindered by the eigendecomposition bottleneck for high-dimensional matrices in the embedding process. In this study, a homotopy continuation algorithm is applied for improving this embedding model by parallelizing the corresponding eigendecomposition. The eigenvalue solution is converted to the operation of ordinary differential equations with initialized values, and all isolated positive eigenvalues and corresponding eigenvectors can be obtained in parallel according to predicting eigenpaths. Experiments on the real data sets show that the homotopy-based approach is potential to be implemented for millions of data sets. [ABSTRACT FROM AUTHOR]

In this paper, the solution of the polynomial system arising from sparse interpolation problems with equally spaced sampling is studied. Exploiting the special structure of the polynomial system, it is proved that all its solutions lie in one orbit for generic data. Then, based on the coefficient parameter homotopy method, an efficient algorithm for solving the system is proposed. Smoothness and accessibility of the coefficient parameter homotopy hold for the constant terms of the start system generated by a randomly chosen starting point. So, in the coefficient parameter homotopy method, it needs no computational cost in the first phase and only one path needs to be traced in the second phase. Preliminary numerical tests show that the proposed algorithm is promising. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional linear and nonlinear integral equations of the second kind is analyzed. The homotopy analysis method (HAM) is used for determining the solution of the investigated equation. In this method, a solution is sought in the series form. It is shown that if this series is convergent, its sum gives the solution of the considered equation. The sufficient condition for the convergence of the series is also presented. Additionally, the error of approximate solution, obtained as partial sum of the series, is estimated. Application of the HAM is illustrated by examples. [ABSTRACT FROM AUTHOR]