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Firstly, a perturbation based correction of an admissible open-loop control in an unperturbed problem is constructed in the form of linear feedback. The constructed control transfers the perturbed system to a neighborhood of a given point. Secondly, then the resulting feedback is refined to reach the end point. The proposed algorithm is illustrated by numerical experiments.

Mathematical modeling of the nonlinear problem biological population nondivergent form with absorption, variable density are considered. Qualitative properties of considered problem which include the critical and double critical cases are discussed. Results of the numerical experiments are analyzed.

The nonlinear auto-regressive moving average with exogeneous input (NARMAX) model known as one of superior type of models to represent a wide class of dynamic systems. In this paper, a modified NARMAX is proposed in modeling dynamic system. The aim is to investigate the performance of the modified NARMAX model and compared to the conventional NARMAX model for modeling CE8 coupled electric drives system. Multi-objective optimization differential evolution (MOODE) algorithm is used as a model structure selection algorithm to obtain the final model from both approached models. Model predicted output (MPO) test is applied in order to reveal the performance of each model. Through the MPO test, it is concluded that the modified NARMAX model offers a better predicted output than conventional NARMAX model.

In this article the solvability of the problem of optimal control of oscillatory processes described by the integro- differential equation with the Fredholm operator, with given control limitation is investigated.It is established that the desired control is among the solutions of the nonlinear Fredholm integral equation of the first kind. Sufficient conditions for the existence of a solution of nonlinear optimization problem are found.

In this paper, we consider a variant of the fictitious domain method associated with the modification of nonlinear terms in a fictitious subdomain. The model problem shows the effectiveness of using this modification. The proposed version of the method is used to solve the problem of an arbitrary region and to set a boundary condition for pressure. A numerical solution is implemented and the results of numerical results are given.

The article explores general nonlinear equations with a parameter. Sufficient conditions for the existence of a solution of nonlinear integral equations in the form of the sum of two functions for the individual values of the parameter are found.

The extended first integral method is used to study the complete propagation characteristics of nonlinear surface plasmon polaritons in a multilayer slab waveguide. The method has an advantage of simultaneously obtaining dispersion relation as well as field profile of the propagating mode in the guiding structure. The dispersion characteristics have been found to depend on the input field intensity. Soliton - like nature for the electric field in the focussing nonlinear medium has been found to enhance the propagation length of the wave propagating through the waveguide.

Haar wavelets have numerous applications in the solution of differential equations. This paper presents the proficient Haar wavelet technique for the numerical solution of the one important class of nonlinear partial differential equation. The implementation of the conventional numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this, we have proposed Haar wavelet based numerical method using operational matrices. The test problem is presented to reveal the applicability and attractiveness of the proposed scheme.

This article reveals the influences of different kinds of chemical reactions such as homogeneous-heterogeneous with heat generation/absorption including non-linear radiation on a two-dimensional steady Newtonian laws obeyed hydromagnetic nanofluid flow through a stretching sheet. In this investigation we consider the magnetic field in the normal direction to the flow. The base fluids are water ( H2O ) and ethylene glycol (C2H6O2) which are contains the nanoparticles of silver (Ag). The partial differential equations of nonlinear type with boundary conditions which illustrates the flow of the problem are solved numerically after using suitable similarity transformations for transforming into ordinary differential equations of nonlinear type. Graphical representations for species concentration, fluid velocity and fluid temperature with influences of physical and fluid parameters. We illustrate various values of τ (skin friction), Nu (Nusselt Number) and Sh (Sherwood number) in the tabular form.

In an experiment, the contribution to the optical nonlinear effects mainly depends upon the characteristics of the pump pulse such as the repetition rate and the pulse duration. Under high repetition rate (HRR) laser excitation, significant thermal effects are also present along with the optical effects even with femtosecond nJ pulse energies. In this work, these effects have been distinguished by the technique of Z-scan by varying the pump pulse duration.

This paper aims to establish some new sufficiency conditions for the oscillatory behavior of the solutions of a forced non linear difference equation of fractional order with damping term Δ(Δμu(l))+γ(l)Δμu(l)+ρ(l)Ψ(u(l))=ϕ(l),l∈ℕ0Δ(μ−1)u(0)=u(0)=u0 with the assistance of properties of Riemann-Liouville (RL) sum and difference operators. Suitable examples are provided to demonstrate the applicability of the derived outcomes.

Optical glasses with high third-order optical nonlinearities are promising for the development of progressive nonlinear photonic devices. In view of this, Eu3+ doped lanthanum lead borate glasses with varied concentrations of PbO have been prepared and their optical nonlinearities were studied by utilizing Z-scan technique. The variation of nonlinear coefficients demonstrated an increase trend as a function of PbO in the glass composition. Hence the studied glasses are believed to be beneficial for photonic applications.

The dispersion plays an important role in pulse propagation in an optical fiber. The effect of higher order dispersion terms are generally neglected for pico second pulse propagation. In this paper we have tried to discuss the propagation length upto which the third order dispersion (TOD) parameter may be neglected. Beyond this length the ignorance of this parameter is not possible as it may affect the pulse shape. Semiconductor Bloch equations are applied to determine the Polarization induced in the medium due to incident Gaussian pulse. The analysis of induced polarization is made by Non linear Schrodinger equation. To represent the pulse propagation behavior linear partial differential equation is used. The results of the numerical calculations provide a support for the fact that the third order dispersion parameter in an optical fiber may be neglected only upto a specified length depending upon the input pulse width.

A comprehensive numerical analysis is carried out to study the effect of Hall current and viscous dissipation on MHD flow over an exponentially accelerated plate with ramped temperature. The emerging flow governing coupled nonlinear dimensional velocity, energy and concentration equations along with appropriate initial and boundary conditions are transformed into non-dimensional form by using suitable transformations. An efficient, well-validated and versatile numerical Finite element method is engaged to find the solution of resulting dimensionless equations. The variations in flow pattern due to the impact of several controlling parameters are illustrated graphically. The values of skin frictions, Nusselt number and Sherwood number are presented in a tabular form against different parameters. Finally current numerical code is validated with the published literature and a great accordance has been achieved.

The analytical solution of vibration on Single wall Carbon nanotubes (SWNTs) embedded in viscous material with linear and nonlinear damping subjected to parametric excitation has been investigated. The analytical non resonant solution of amplitude of nonlinear vibration is deduced using Krylov Bogoliubov and Mitropolsky method. The solution is determined till second order of approximation.

Concepts of q-deformation have been extended to numbers in recent past and q-deformed nonlinear maps have been introduced in this context. Its dynamics have been extensively investigated in logistic and Gaussian maps. These systems display multistability and coexisting attractors which are usually absent in one-dimensional system. We extend this definition to tent maps. We also propose a scheme of diffusively coupled q-deformed logistic and tent maps. We study the bifurcation diagrams in these systems. A detailed study of dynamical phase portraits of these systems for various system parameters. We find that multistability found in the single map is absent in the coupled system in the thermodynamic limit. For q-deformed tent map or coupled q-deformed tent maps, there are no periodic or even band-periodic windows or coexisting attractors. In this sense, we observe robust chaos. However, for coupled q-deformed tent maps the accessible phase space area is somewhat reduced.

PI and PID controllers are usually found to provide poor performance for high-order, nonlinear processes, and multiple-input and multiple-output (MIMO) systems with strong loop interactions. One of the most common multivariable system found is the distillation column. In this study, a new structure of fuzzy self-tuning PI (fuzzy-PI) controller for distillation column is proposed. The values of Kp and Ki are initially tuned with Ziegler-Nichols closed loop method and are always updated based on the system's condition. With the self-tuning structure, the fuzzy-PI controller improves the conventional PI controller in terms of the linear distillation column control performance. The controllers’ performance are tested by XD and XB set point change and XF disturbance change.

The equations of motion of the turbocharger supported on fluid film bearings have been developed, considering the rotor shaft rigid and flexible, separately. To solve the nonlinear equations of motion of the system, MATLAB® routine ode15s solver has been adopted, and system response is calculated. The behavior of the system has been studied using orbital plots and phase portraits. The Floquet theory has been used to investigate the stability of the system. The Floquet multipliers show that the rigid and flexible rotor will place a considerable influence on the periodic solution of the system.

This work considers discrete fractional nonlinear equations with Riemann-Liouville (R-L) type difference operator of the form Δβ[v(l)−p(l,v(l))]=f(l+β,v(l+β)),l∈N0,0

This paper presents a simple and efficient 2D corotational formulation for geometrically nonlinear dynamic analysis of beam structures. Newmark’s implicit time-stepping scheme with the iterative predictor-corrector algorithm is employed along with the energy-convergent criterion to determine the nonlinear dynamic equilibrium stage. The validity and efficiency of the proposed formulation are verified by simulating large deformation of a swinging rubber beam. This study confirms that the proposed formulation is highly accurate and efficient. Despite the simple numerical implementation, the proposed method is reliable in performing simulations of flexible beam structures undergoing extremely large deformation or even a chaotic motion.

In the present study, we report the preparation of ZnO nanoparticles and their linear and nonlinear optical (NLO) properties. The ZnO nanoparticles are prepared by chemical route method. The linear optical studies such as UV-Vis absorption and photoluminescence show the strong absorption at 333 nm and emission at 450 nm, respectively. The direct ban near optical (NLO) properties are investigated using Z-scan technique under DPSS CW laser at 532 nm wavelength. Mean time strong nonlinear absorption and negative nonlinear refractions are observed. The third order nonlinear optical coefficients and the magnitude of the susceptibility are determined. The optical limiting behavior of the nanoparticles is observed and the limiting threshold is calculated. The above results show that the prepared ZnO nanoparticles are suitable for nonlinear optical and optoelectronic applications.

This article analyses the heat transfer effect of melting on magneto-hydrodynamic Casson fluid flow surrounded by porous medium in the presence of first order chemical reaction with radiation. By using appropriate similarity transformations the governing partial differential equations are converted into ordinary differential equations and then solved the obtained nonlinear equations by using Runge-Kutta algorithm supported by shooting technique. A comprehensive analysis of certain parameters on the velocity, temperature, concentration with the support of graphs. Coefficient skin friction and reduced Nusselt number are explained using graphs..

In this paper, we want to study the different kinds of solutions of second order nonlinear non-homogeneous neutral delay dynamic equation with mixed type, classifying these solutions into four classes and obtained conditions for the existence/non-existence of solutions in these classes. Also we have mentioned examples to an important results.

Present article is devoted to the work on two-dimensional EMHD flow of Carreau fluid across a stretching sheet within the sight of thermal radiation, electric field parameter. We assumed convective boundary condition for this problem. Leading equations (partial differential equations) of this problem have been transmuted as ordinary differential equations (nonlinear) by using similarity transmutations and then solved mathematically by expending shooting technique. Plots are used to display the influence of the pertinent parameters including electric field parameter on the regular profiles (temperature and velocity). And also, we have discussed the influence of the same parameters on friction factor and Nusselt number using table. It is seen that electric field parameter is useful to elevate temperature. And also, we noticed that the magnetic field parameter lowers the rate of heat transfer.

This paper numerically investigates magnetohydrodynamic, dissipative and mixed convection boundary layer copper-water nanofluid fluid flow over a nonlinear stretching/shrinking sheet in the existence of heat generation/absorption and viscous dissipation. The governing coupled nonlinear momentum and thermal energy boundary layer equations are transformed to an ordinary differential equation by using local similarity transformations and then an implicit finite difference scheme is espoused to solve the nonlinear differential equations. The pertinent physical parameters which governs the flow problem are the magnetic field, Richardson number, power law stretching parameter, suction/injection parameter, Eckert number and heat source/sink parameter influenced the velocity and temperature distributions and are portrayed with the aid of graphs. The skin friction coefficient and Nusselt number values are also presented and discussed. The temperature profile increased with an increasing value of and for together stretching and shrinking sheet cases.

Motivated by an important role of growth kinetics of particulate ensembles in metastable liquids, we develop the theory of nucleation and growth of spherical crystals in a binary supercooled melt at the intermediate stage of a phase transition process. An integro-differential set of governing equations consisting of the Fokker-Planck and balance equations with allowance for the nonlinear growth and nucleation kinetics is formulated and analyzed. The particle population density distribution function and the degree of system metastability are found analytically with allowance for fluctuations of newly born crystals in a metastable binary liquid. We demonstrate that the nonlinear kinetics of crystal growth plays a decisive role in the nonlinear dynamics of particulate ensembles.

A mathematical model of heat transfer in the field of forest combustible material (FCM) with a water barrage was formulated. The model describes the evaporation of moisture at the interface between the dry and wet parts of the FCM, as well as the radiant heat exchange between a high-temperature heating source (flame) and the surface of the material. The solution of the nonlinear nonstationary differential equation in partial with the appropriate boundary conditions was carried out using the control volume method. Based on the results of numerical modeling, a conclusion about the possibility of localizing forest fires using barrage of moisture-saturated forest combustible material was drawn.

Friction is a phenomenon which occurs commonly in the nature and in mechanical constructions. One can find numerous mathematical models describing friction as a one-dimensional process. On the other hand, the number of models which take into account second dimension is significantly smaller. The paper in hand introduce two-dimensional, dynamical model for orthotropic dry friction. Proposed model obeys maximum dissipation power principle by means of non-linear two-dimensional bristle stiffness. Numerical studies show influence of orthotropic friction on planar oscillator and 2D stick-slip system trajectories. Model is also verified against experimental results. Frictional pair with orthotropic properties have been prepared for laboratory rig which is a spherical pendulum with frictional contact. Comparison between experimental and simulation results shows good similarity, although further validation is required.Friction is a phenomenon which occurs commonly in the nature and in mechanical constructions. One can find numerous mathematical models describing friction as a one-dimensional process. On the other hand, the number of models which take into account second dimension is significantly smaller. The paper in hand introduce two-dimensional, dynamical model for orthotropic dry friction. Proposed model obeys maximum dissipation power principle by means of non-linear two-dimensional bristle stiffness. Numerical studies show influence of orthotropic friction on planar oscillator and 2D stick-slip system trajectories. Model is also verified against experimental results. Frictional pair with orthotropic properties have been prepared for laboratory rig which is a spherical pendulum with frictional contact. Comparison between experimental and simulation results shows good similarity, although further validation is required.

In this paper, we proposed and analyzed the Omnivore -predator-prey model with the II-Holling functional response to the interaction between Predator-Prey and the Nonlinear functional response to the interaction between Omnivore –prey, existence, uniqueness, and boundedness of solution of the system are studied. All equilibria points in the system coexist. local stability and local bifurcation are investigated around each equilibria points. Finally, numerical simulations are used to investigate theoretical results.

Finite element (FE) analyses of reinforced concrete (RC) structures or structural elements using nonlinear concrete material models are currently a subject of interest for many scientific institutions or departments. Obtaining of relevant results from such analyses depends on a number of aspects, including (for example) the level of complexity of the computational model used, correct boundary conditions, the correct definition of parameter values of the material models used, etc. A key aspect from the above-mentioned aspects is especially the problematic of the definition of correct parameter values of the used material models. Defining the correct parameter values of the material models is often a very difficult task because we usually do not have enough knowledges about the given parameters or their possible ranges. However, the current level of computational technology combined with the current state of knowledge in the fields of mathematical statistics, artificial neural networks and optimization allows us to successfully deal with the given task by so called inverse identification of parameters whose correct numerical values we need to obtain. The aim of this paper is to show the possibility of using the optimization-based inverse identification of parameter values of a nonlinear concrete material model for FE analysis of structural element which is a high RC beam subjected to a four-point bending. For this purpose, the results of the corresponding experiment are used.

In the stationary regime, configurations of corrected Kompaneets equation have been analyzed within a theoretically approach and the spectrum evolution for the scattered photons has been derived. A particular attention has been granted to the impact of the nonlinearity on the spectrum distorsions, mainly to the closed form special solutions expressed in terms of Heun functions. In the nonrelativistic limit, stationary solutions of the Kompaneets equation bring into attention the Heun double confluent functions. On the other side, when various type of corrections come into play, the photon distribution functions are governed by the Heun triconfluent functions. In magnetar’s magnetosphere, where the resonant Compton scattering is more important than the Compton one, the photon number density can be written in terms of Hermite and confluent hypergeometric functions.In the stationary regime, configurations of corrected Kompaneets equation have been analyzed within a theoretically approach and the spectrum evolution for the scattered photons has been derived. A particular attention has been granted to the impact of the nonlinearity on the spectrum distorsions, mainly to the closed form special solutions expressed in terms of Heun functions. In the nonrelativistic limit, stationary solutions of the Kompaneets equation bring into attention the Heun double confluent functions. On the other side, when various type of corrections come into play, the photon distribution functions are governed by the Heun triconfluent functions. In magnetar’s magnetosphere, where the resonant Compton scattering is more important than the Compton one, the photon number density can be written in terms of Hermite and confluent hypergeometric functions.

It is well known that microcracks generate strong higher harmonics in propagating monochromatic waves. There is a large amount of literature on modeling this phenomenon, but most of these existing papers only describe one specific mechanism. For example, Zhao et al. [1] assumes that the crack faces are either open under tension or closed under compression, and in the latter case they may slide against each other. On the other hand, the Nazarov and Sutin [2] model assumes microcracks as an elastic contact of two rough surfaces, which are never completely separated by an external load. All these mechanisms depend on the level of excitation. In this research, a micromechanical model for the acoustic nonlinearity generation of microcracks is developed by combining the bilinear stiffness model and the rough surface contact model to describe the excitation-dependent nonlinear behavior of distributed microcracks. It is shown that the first and second harmonic amplitudes have the relationship: A2~A1n, with n dependent on the amplitude of excitation, and 2 ≥ n ≥ 1 for non-adhesive crack surfaces. Nanostructured ferritic alloys (NFAs) [3] are considered as an example. These materials exhibit outstanding high-temperature properties, irradiation tolerance and thermal stability, making them a leading candidate for advanced nuclear fission and fusion applications. One characteristic property of mechanically processed NFAs is their layer-like structure, with a large number of microcracks aligned in a specific direction. Nonlinear ultrasound measurements (acoustic nonlinearity, β) with longitudinal waves are used to characterize this material. The results show that these measurement techniques are sensitive to the orientation of the cracks. The model developed in this research is then used to interpret these experimental measurements and used to characterize the microcracks in a NFA specimen.

The split-step modified cubic B-splines collocation (SSM3BC) method is constructed by combining the split-step approach with the modified cubic B-splines collocation method for the two-dimensional (2D) nonlinear Schrodinger (NLS) equation. Numerical tests are carried out, and the SSM3BC schemes are efficient both for the one-dimensional (1D) and 2D NLS equations. It is also numerically verified that the proposed 1D and 2D schemes are second-order convergence both in time and space.

Nondestructive evaluation techniques have been used to evaluate the integrity of bolted joints in structures. One such method is nonlinear wave modulation spectroscopy (NWMS), which examines the sideband generation in the presence of dual inputs as a result of nonlinearity present in the system. This work uses NWMS on an aluminum T-joint structure to evaluate the torque levels of the bolts. In addition, this work compares excitation with two shakers to excitation with a single shaker and an impact hammer, in which the hammer excites several low frequency modes. The results show that the damage indicator based on the sidebands is highly dependent on damage and sensor location, and that the damage indicator is most sensitive over a specific range of torque levels.Nondestructive evaluation techniques have been used to evaluate the integrity of bolted joints in structures. One such method is nonlinear wave modulation spectroscopy (NWMS), which examines the sideband generation in the presence of dual inputs as a result of nonlinearity present in the system. This work uses NWMS on an aluminum T-joint structure to evaluate the torque levels of the bolts. In addition, this work compares excitation with two shakers to excitation with a single shaker and an impact hammer, in which the hammer excites several low frequency modes. The results show that the damage indicator based on the sidebands is highly dependent on damage and sensor location, and that the damage indicator is most sensitive over a specific range of torque levels.

One of the methods to mitigate fouling is to determine the optimal geometries of the heat exchanger. In this research, optimization of heat exchanger design was performed by determining shell and tube heat exchanger geometries. In this optimization require three components i.e. fouling and heat exchanger model, problem formulation and optimization technique. Fouling model was built by utilizing Polley fouling model. The objective function of optimization minimizes fouling resistance by determining shell and tube heat exchanger geometries, with constraints available heat duty and allowable pressure drops. Since the optimization falls into mixed-integer nonlinear problems that have some local optimum, the optimization technique has the capability to find out the global optimum is required. The selected optimization variables are shell side diameter (Ds), tube outside diameter (do) and the number of baffles (Nb). Optimization results showed that shell side fouling resistance was reduced by 17%, tube side was reduced by 48% compared with existing design by considering available heat duty and allowable pressure drops.

This paper is devoted to the leader-following problem involving a Caputo-type fractional derivative with respect to another function, which is a mathematical model describing dynamics of the leader and a number of agents. The sufficient conditions are given to guarantee that the leader-following consensus is achieved in the system. The main tools in the investigation are the properties of Volterra equations and a standard fixed point theorem.This paper is devoted to the leader-following problem involving a Caputo-type fractional derivative with respect to another function, which is a mathematical model describing dynamics of the leader and a number of agents. The sufficient conditions are given to guarantee that the leader-following consensus is achieved in the system. The main tools in the investigation are the properties of Volterra equations and a standard fixed point theorem.

In this work, we shall be concerned with the existence result to the nonlinear elliptic equations −div(a(x,u,∇u))+Φ(x,u))+g(x,u,∇u)=0 in the setting of Orlicz spaces. The results obtained are proved using the sub- and supersolution method.

For the design problem of special clothing for high temperature operation, firstly, the heat transfer process of each layer under unstable state is analyzed, the corresponding temperature distribution is obtained, and the heat transfer analysis model is established to solve the problem. Second, because each layer has different materials under non-steady state, we have established different initial conditions and boundary conditions for each layer of fabric. Finally, the fake body wearing protective clothing is abstracted into a spherical model to establish a three-dimensional heat conduction model of fabric and fabric, fabric and air, human body and human body. The directed differential equation method is used to solve the set of partial differential equations, and the explicit implicit method is used to solve the nonlinearity of the partial differential equations, and the temperature distribution of each layer is obtained.

This research applies nonlinear ultrasonic techniques for the quantitative characterization of additively manufactured materials. The characterization focuses on identifying the dislocation density produced during the additive constructive process in order to increase confidence on a part’s performance and the success of the manufacturing process. Second harmonic generation techniques based on the transmission of Rayleigh surface waves are used to measure the ultrasonic nonlinearity parameter, β, which has proven a quantitative indicator of dislocations but has not been fully proven in additive manufactured materials. 316L and 304L stainless steel parts made from Powder Bed Fusion and Laser Engineered Net Shaping are compared between AM techniques and with wrought manufactured counterparts. β is consistently higher for additive manufactured parts. An annealing heat treatment is applied to each specimen to reduce dislocation density. β expectedly decreases by annealing in all specimens. A linear ultrasonic measurement is made to evaluate the effectiveness of using nonlinear techniques. The ultrasonic attenuation is higher for additive manufactured parts and increases at higher frequencies.

The problem of proposing a framework for linear moment matching is tackled. It is possible to define notions of moments, i.e., from a complex "s-domain" approach and from a time-domain point of view. Based on these notions, reduced order models that achieve moment matching are obtained. This paper sets the basis to develop the theory of model reduction by moment matching for PDE (nonlinear) systems.The problem of proposing a framework for linear moment matching is tackled. It is possible to define notions of moments, i.e., from a complex "s-domain" approach and from a time-domain point of view. Based on these notions, reduced order models that achieve moment matching are obtained. This paper sets the basis to develop the theory of model reduction by moment matching for PDE (nonlinear) systems.

This article is devoted to the detailed study of one of the models obtained by group analysis in [P. Voraka and S. V. Meleshko. Group classification of one-dimensional equations of fluids with internal energy depending on the density and the gradient of the density. Commun Nonlinear Sci Numer Simulat 14 (2009) 2111-2119]. The main feature of this model is the equations describing the motion of fluids with the internal energy function Cργαp + C1ρµ. Optimal systems of subalgebras are constructed. Representations of all invariant solutions are obtained.

Reduction Kiln plays a very important role in the nickel smelting process. As multi-variable processing and nonlinear system, nickel kiln is difficult to be maintain in the best performance. While research on this area is very rare for this industry is such specific and unique. In this paper, it is proposed time-series neural network model for nickel kiln, with an approach based on Nonlinear Auto Regressive with eXogenous input (NARX). This model provides prediction of input-output data set to achieve certain operating condition, especially profile temperature alongside the kiln. Input-output data set of real plant was collected from Kiln 5 in PT Vale Indonesia for along 2018. The data set was reviewed and chosen for the neural network model. It was achieved that regression of the model is 0.999 which is proving that there is strong relationship between input and output. While mean squared error (MSE) is 16.4077, which means that the model is able to capture the characteristic of the plant and temperature of 6 points alongside the kiln, yet open for future research.

The aim of this paper is to study mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple decision variables without any constraints. We propose the Gradient Method (Steepest Descent Method of Cauchy) for solving unconstrained multi-variable fuzzy optimization problem using H-differentiability of fuzzy valued functions. A numerical example is provided to illustrate the proposed method.