Solutions Of Nonlinear Equations Research Articles

Analytical and Approximate Solutions of the Nonlinear Gas Dynamic Equation Using a Hybrid Approach

This paper presents the study of a numerical scheme for the analytical solution of nonlinear gas dynamic equation. We use the idea of Laplace–Carson transform and associate it with the homotopy perturbation method (HPM) for obtaining the series solution of the equation. We show that this hybrid approach is excellent in agreement and converges to the exact solution very smoothly. Further, HPM combined with He’s polynomial is utilized to minimize the numerical simulations in nonlinear conditions that make it easy for the implementation of Laplace–Carson transform. We also exhibit a few graphical solutions to indicate that this approach is extremely reliable and convenient for linear and nonlinear challenges.

Unique iterative solution for high-order nonlinear fractional q-difference equation based on psi -(h,r)-concave operators

An objective of this paper is to investigate the boundary value problem of a high-order nonlinear fractional q-difference equation. It was to obtain a unique iterative solution for this problem by means of applying a novel fixed-point theorem of psi -(h,r)-concave operator, in which the operator is increasing and defined in ordered sets. Moreover, we construct a monotone explicit iterative scheme to approximate the unique solution. Finally, we give an example to illustrate the use of the main result.

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

The classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

Kashuri Fundo Decomposition Method for Solving Michaelis-Menten Nonlinear Biochemical Reaction Model

In most of the real life problems, we encounter with nonlinear differential equations. Problems are made more understandable by modeling them with these equations. In this way, it becomes easier to interpret the problems and reach the results. In 1913, the basic enzymatic reaction model introduced by Michaelis and Menten to describe enzyme processes is an example of nonlinear differential equation. This model is the one of the simplest and best-known approaches of the mechanisms used to model enzyme-catalyzed reactions and is the most studied. For most nonlinear differential equations, it is very difficult to get an analytical solution. For this reason, various studies have been carried out to find approximate solutions to such equations. Among these studies, those in which two different methods are used by blending attract attention. In this study, a blended form of the Kashuri Fundo transform method and the Adomian decomposition method, so-called the Kashuri Fundo decomposition method, is used to find a solution to the Michaelis-Menten nonlinear biochemical reaction model in this way. This method has been applied to the biochemical reaction model and an approximate solution has been obtained for this model without complex calculations. This shows that the hybrid method is an effective, reliable, simpler and time-saving method in reaching the solutions of nonlinear differential equations.

Heat and mass transfer in the processes of drying thin natural leathers pasted on smooth surfaces

The results of the study of drying thin natural leather pasted on a smooth surface are presented. The results of solving the differential equation of non-stationary heat conduction with constant thermophysical coefficients are used to determine the average temperature during the period of decreasing drying rate. The calculated values of temperatures for moisture-proof and moisture-free skin surfaces are given. Methods for simplifying solutions of nonlinear equations with the aim of linearizing these equations for approximating solutions with variable transport coefficients are considered. The use of the method of piecewise stepwise approximation of the transfer coefficients with constant values of these coefficients over short time intervals showed that for low-intensity drying processes, the solutions of the linear heat equation are implemented quite accurately, confirming the regularities obtained empirically. The calculation of the duration of drying according to the method of B. S. Sazhin is given. The results of the processing of experimental curves for drying leathers pasted are presented. The reliability of the obtained equations is verified and the experiment is compared with the calculated values using the formulas. The obtained approximate analytical solutions with reliable values of the transfer coefficients, confirmed by experimentally established regularities, are of practical importance. Together with experimental methods, analytical methods make it possible to establish optimal drying regimes and more accurately generalize experimental data.

An analytical method for finding exact solutions of a nonlinear partial differential equation arising in electrical engineering

In this investigation, we aim to investigate the novel exact solutions of nonlinear partial differential equations (NLPDEs) arising in electrical engineering via the -expansion method. New acquired solutions are kink, particular kink, bright, dark, periodic combined-dark bright and combined-dark singular solitons, and hyperbolic functions solutions. The achieved distinct types of solitons solutions contain critical applications in engineering and physics. Numerous novel structures (3D, contour, and density plots) are also designed by taking the appropriate values of involved parameters. These solutions express the wave show of the governing models, actually.

New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method

Abstract In this article, first integral method (FIM) is used to acquire the analytical solutions of (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equation. New soliton solutions are obtained, such as solitons, cuspon, and periodic solutions. FIM is a direct method to acquire soliton solutions of nonlinear partial differential equations (PDEs). The proposed technique can be used for solving higher dimensional PDEs. FIM can be implemented to solve integrable and ion-integrable equations.

Generalized differential transformation method for solving two-interval Weber equation subject to transmission conditions

The main goal of this study is to adapt the classical differential transformation method to solve new types of boundary value problems. The advantage of this method lies in its simplicity, since there is no need for discretization, perturbation or linearization of the differential equation being solved. It is an efficient technique for obtaining series solution for both linear and nonlinear differential equations and differs from the classical Taylor’s series method, which requires the calculation of the values of higher derivatives of given function. It is known that the differential transformation method is designed for solving single interval problems and it is not clear how to apply it to many-interval problems. In this paper we have adapted the classical differential transformation method for solving boundary value problems for two-interval differential equations. To substantiate the proposed new technique, a boundary value problem was solved for the Weber equation given on two non-intersecting segments with a common end, on which the left and right solutions were connected by two additional transmission conditions.

Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a $$C_0$$ -semigroup $$W(\cdot )$$ generated by the operator $$-A$$ and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding $$C_0$$ -semigroup $$W(\cdot )$$ and non-instantaneous impulsive function $$\gamma _k$$ are compact, $$W(\cdot )$$ is not compact and $$\gamma _k$$ is compact, $$W(\cdot )$$ and $$\gamma _k$$ are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given.

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html

The Extended Rayleigh–Ritz Method for Higher Order Approximate Solutions of Nonlinear Vibration Equations

The Extended Rayleigh–Ritz Method for Higher Order Approximate Solutions of Nonlinear Vibration Equations

Computational finite element analysis of electromagnetic radiative nanofluid containing motile germs with chemical reactive process

In this paper, to characterize the appearances of bio-convection and influencing microorganisms in movements of influenced Walters-B nanofluid, we built a process in a stretchable sheet utilizing Fourier’s and Fick’s laws. The mass and temperature diffusion hypothesis proposed by Cattaneo and Christov has been considered. Also, the Buongiorno phenomenon for nanofluid wave in the Walters-B fluid is used, according to the Cattaneo–Christov connection. A system of regulating partial differential equations (PDEs) is transformed into ordinary differential equations (ODEs) using similarity transformations. Due to the Galerkin finite element mode, the software COMSOL is utilized to derive mathematical solutions for nonlinear equations (G-FEM). Temperature, velocity, and microorganism analysis are all investigated using logical and pictorial metrics. Nanoparticle concentrations are also calculated by considering various approximations of well-known physical characteristics. The simulations of this model demonstrate a straight association between the heat field and the current Biot number and factor of the Walters-B liquid. The temperature field is improved to increase the thermal Biot amount and factor approximations for the Walters-B fluid.

Trigonometric Identities and Entire Solutions of Non-linear Binomial Differential Equations

Trigonometric Identities and Entire Solutions of Non-linear Binomial Differential Equations

Strong convergence of the tamed Euler method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments

This paper develops the tamed Euler method (TEM) for approximating solutions of nonlinear hybrid stochastic differential equations with piecewise continuous arguments (SDEPCAs). As far as we know, there are only two published articles (Hou et al., 2009; Zhang, 2020) considered the Lp-convergence rate of numerical methods for hybrid systems, and the rates are no more than 1/p(p≥2) in both of these two studies. In this work, we use the technique proposed in Song et al. (2022) to develop the TEM, and obtain the following two main results: (i) The TEM converges strongly to the exact solution of hybrid SDEPCAs; (ii) The Lp-convergence rate (p≥2) can reach 1/2. Finally, we give two numerical examples to verify the theoretical results.

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

AbstractWe establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz–Sobolev spaces and under general growth conditions on the convection term. The sub‐ and supersolutions method is a key tool in the proof of the existence results.

Extended Convergence of Two Multi-Step Iterative Methods

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

Local Convergence of Parameter Based Derivative Free Continuation Method for the Solution of Non-linear Equations

This paper's major purpose is to evaluate the local convergence of the parameter-based sixth- and seventh-order continuation iterative approach for solving nonlinear equations in R. This analysis assumes that the Fréchet derivative of the first order satisfies the Lipschitz continuity condition. Under these circumstances, we explore convergence analysis in order to investigate the existence and uniqueness region for the solution of our proposed strategies. Thus, we also offered the theoretical concept of the radii of convergence balls for the proposed approach. By determining the radii of the convergence balls and solving many numerical problems, we can verify the significance of our convergence study.

Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

The Fibonacci wavelets approach for the fractional Rosenau–Hyman equations

In this study, a novel approach, called the Fibonacci wavelet collocation technique (FWCT), is presented for the numerical solution of nonlinear fractional order partial differential equations (FPDEs). The chosen numerical strategy is based on the collocation method with the Fibonacci wavelet and its functional integration matrix. The newly developed numerical approach for the nonlinear time-fractional Rosenau–Hyman equation is the focus of this paper. The corresponding fractional order derivative is taken in the Caputo sense. This effective numerical approach converts the nonlinear problem into a set of algebraic-type nonlinear equations. This nonlinear system of equations is analyzed using the Newton–Raphson method to find the unknown coefficients. We provide the different types of error analysis of the present technique. The exact solution and the results obtained by the other literary methods are compared to the results achieved. The numerical outcomes demonstrate that the current analysis offers a more accurate approximation than the prior approaches. The result derived with the aid of this technique is in good agreement and indicates that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally attractive, efficient, effective, reliable, and robust for solving various physical models in science and engineering. We made all calculations and plotted the graphs using Mathematica software.

Rational-type soft fuzzy contraction in soft fuzzy metric space and solution of nonlinear integral equation

This paper is devoted to the explication of the introduction of rational-type soft fuzzy contraction in soft fuzzy metric space. New fixed point results for these new contractions along with relevant and valid examples have been presented and generalized. We have also presented an application to our new formed results in finding the solution of a nonlinear integral equation.