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High-order continuum models are needed for properly capturing the post-failure mechanical responses of soils involving shear bands. Through analysis on the evolution of shear band in granular soils based on a previously proposed micropolar hypoplastic model, a governing equation for the shear band in the critical state was obtained, which is a special nonlinear ordinary differential equation satisfied by the Cosserat angular velocity. A concise derivation of the governing equation was conducted. The properties of the governing equation, the range of the chief parameter and the approach to the solution were mainly discussed. An energy balance equation was formulated as a complementary condition for the determinant of the problem through analysis on the mechanical properties of the shear band. Then, the complete solutions, including the shear-band thickness factor, the stress distribution, the strain rate components, and the shear velocity, were obtained through numerical integration. The shear band thickness factor is particularly useful in determination of the micro-strength parameter of the constitutive model. [ABSTRACT FROM AUTHOR]

We consider an initial value problem for a second-order nonlinear dynamical system that allows discontinuous inputs. In this problem, one state variable is restricted in a closed and bounded interval while the other state variable is unlimited. The existence and uniqueness of the solution to this problem is studied in a time interval determined by two adjacent instants at which the input may be discontinuous. Some results are obtained on the maximal interval of existence of the unique solution in this time interval. The results are then extended to the entire time domain. A robot example is given to verify our results. [ABSTRACT FROM AUTHOR]

Since the droughts of the 1970s–1980s, populations in the Sahel region have opted for a mass exodus to more humid urban or rural centers. Migrations or exoduses have accelerated in recent decades due to environmental degradation and unfavorable climatic conditions. Insufficient harvests are the main reason for migration for the majority of migrants in the Sahelian areas. Migration is a major adaptation strategy to cope with extreme climatic conditions, thus requiring quantification in the destination area. The aim of this paper is to propose a metapopulation model to approximate reality by identifying the transition from one socioeconomic vulnerability group to another, from a less favorable area to favorable area in terms of natural resources, depending on the strategies, policies, and climate variability. The model was used to analyze the dynamics of socioeconomic vulnerability to study the impact of migration on the dynamics of socioeconomic vulnerability. The developed mathematical model was analyzed. Up to 2050, simulations applied to the Tougou village in northern Burkina Faso show that migration has a positive impact on the socioeconomic vulnerability of the destination area, thereby reducing the vulnerability of the population by 10% when resources are increased by up to 30%. [ABSTRACT FROM AUTHOR]

The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein's famous equation m = m 0 1 − v 2 / c 2 , where m 0 is the rest mass of the projectile and c is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to c. It is revealed that the mass of the RP varies during the motion and an analytic formula for the instantaneous mass in terms of time is derived. Also, it is declared that the angle of maximum range of the RP depends on the launching velocity, i.e., unlike the NRP in which the angle of maximum range is always π / 4 . In addition, this angle lies in a certain interval [ π / 4 , π / 6) for any given initial velocity (

In this investigation, we utilize advanced versions of the Extended Direct Algebraic Method (EDAM), namely the modified EDAM (mEDAM) and r+mEDAM, to explore families of optical soliton solutions in the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model (FPRKLM). Our study stands out due to its in-depth investigation and the identification of multiple localized and stable soliton families, illuminating their complex behavior. We offer visual validation via carefully designed 3D graphics that capture the complex behaviors of these solitons. The implications of our research extend to fiber optics, communication systems, and nonlinear optics, with the potential for driving developments in optical devices and information processing technologies. This study conveys an important contribution to the field of nonlinear optics, paving the way for future advancements and a greater comprehension of optical solitons and their applications. [ABSTRACT FROM AUTHOR]

In this article, the flux-charge analysis method is applied to obtain the theoretical response of the voltage generated in a parallel Memristor–Capacitor (M–C) circuit excited by an input pulse generator with a 100 kHz frequency, 5 V amplitude and a 50 ohms output impedance. The theoretical solution of the nonlinear ordinary differential equation that results when applying the method is reached by a numerical method. As a memristive circuit, a previously reported floating memristor emulator was used. The response obtained is compared with the experimental response, generating evidence that the applied analysis method yields an acceptable margin of error with regards to the experimental results obtained, contrasting with other similar reports, where the analyzes are based on theoretical memristive models, and show simulation results only. Summary, the paper would contribute to the analysis and experimental verification of the parallel M–C circuit subjected to a real switched exciting source, using a memristance equation established in an emulator that is different from the equations commonly used in the literature.

In this paper, we developed a new variational iteration method using the quasilinearization method and Adomian polynomial to solve nonlinear differential equations. The convergence analysis of our new method is also discussed under the Lipschitz continuity condition in Banach space. Some application problems are included to test the efficacy of our proposed method. The behavior of the method is investigated for different values of parameter t. This is a powerful technique for solving a large number of nonlinear problems. Comparisons of our technique were made with the available exact solution and existing methods to examine the applicability and efficiency of our approach. The outcome revealed that the proposed method is easy to apply and converges to the solution very fast. [ABSTRACT FROM AUTHOR]

Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.

A class of problems of wave propagation in waveguides consisting of one or several layers that are characterized by linear variation of the squared refractive index along the normal to the interfaces between them is considered in this paper. In various problems arising in practical applications, it is necessary to efficiently solve the dispersion relations for such waveguides in order to compute horizontal wavenumbers for different frequencies. Such relations are transcendental equations written in terms of Airy functions, and their numerical solutions may require significant computational effort. A procedure that allows one to reduce a dispersion relation to an ordinary differential equation written in terms of elementary functions exclusively is proposed. The proposed technique is illustrated on two cases of waveguides with both compact and non-compact cross-sections. The developed reduction method can be used in applications such as geoacoustic inversion. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of a non-negative monotonically increasing solution of the boundary value problem (BVP) H ′ ′ (η) - H p (η) H ′ (η) + η 2 H ′ (η) + 1 2 p H (η) = 0 , η > 0 , H (0) = 0 , lim η → + ∞ η - 1 p H (η) = 1 , using a shooting argument for p > 1 . This BVP arises when we construct the large time asymptotic solution of the modified Burgers equation, using the method of matched asymptotic expansions, on the quarter plane. [ABSTRACT FROM AUTHOR]

The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein’s famous equation m=m01−v2/c2, where m0 is the rest mass of the projectile and c is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to c. It is revealed that the mass of the RP varies during the motion and an analytic formula for the instantaneous mass in terms of time is derived. Also, it is declared that the angle of maximum range of the RP depends on the launching velocity, i.e., unlike the NRP in which the angle of maximum range is always π/4. In addition, this angle lies in a certain interval [π/4,π/6) for any given initial velocity (

In this investigation, we utilize advanced versions of the Extended Direct Algebraic Method (EDAM), namely the modified EDAM (mEDAM) and r+mEDAM, to explore families of optical soliton solutions in the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model (FPRKLM). Our study stands out due to its in-depth investigation and the identification of multiple localized and stable soliton families, illuminating their complex behavior. We offer visual validation via carefully designed 3D graphics that capture the complex behaviors of these solitons. The implications of our research extend to fiber optics, communication systems, and nonlinear optics, with the potential for driving developments in optical devices and information processing technologies. This study conveys an important contribution to the field of nonlinear optics, paving the way for future advancements and a greater comprehension of optical solitons and their applications.

In this article, we define a p-generalized modified error function as the solution to a non-linear ordinary differential equation of second order, with a Robin type boundary condition at x=0. We prove existence and uniqueness of a non-negative $C^{\infty}$ solution by using a fixed point argument. We show that the p-generalized modified error function converges to the p-modified error function defined as the solution to a similar problem with a Dirichlet boundary condition. In both problems, for p=1, the generalized modified error function and the modified error function are recovered. In addition, we analyze the existence and uniqueness of solution to a problem with a Neumann boundary condition.

In this paper, we developed a new variational iteration method using the quasilinearization method and Adomian polynomial to solve nonlinear differential equations. The convergence analysis of our new method is also discussed under the Lipschitz continuity condition in Banach space. Some application problems are included to test the efficacy of our proposed method. The behavior of the method is investigated for different values of parameter t. This is a powerful technique for solving a large number of nonlinear problems. Comparisons of our technique were made with the available exact solution and existing methods to examine the applicability and efficiency of our approach. The outcome revealed that the proposed method is easy to apply and converges to the solution very fast.

In this paper, a hybrid method based on a Legendre wavelet spectral collocation scheme and rational approximation using Legendre rational functions is proposed. This method is utilized to solve nonlinear differential equations on semi-infinite domains. Although Legendre wavelet collocation method possesses spectral accuracy, it is not convenient to implement it on large intervals. On the other hand, Legendre rational functions are useful for approximating the solution of differential equations on large domains whilst this process is computationally cheap. The proposed hybrid method enjoys the benefits of both methods while improves their performance. The proposed method is compared with other numerical methods in the literature to demonstrate its superiority. [ABSTRACT FROM AUTHOR]

A class of problems of wave propagation in waveguides consisting of one or several layers that are characterized by linear variation of the squared refractive index along the normal to the interfaces between them is considered in this paper. In various problems arising in practical applications, it is necessary to efficiently solve the dispersion relations for such waveguides in order to compute horizontal wavenumbers for different frequencies. Such relations are transcendental equations written in terms of Airy functions, and their numerical solutions may require significant computational effort. A procedure that allows one to reduce a dispersion relation to an ordinary differential equation written in terms of elementary functions exclusively is proposed. The proposed technique is illustrated on two cases of waveguides with both compact and non-compact cross-sections. The developed reduction method can be used in applications such as geoacoustic inversion.

In this study, an analytic method based on the Jacobi elliptic functions has been presented to obtain the exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation. This equation is reduced to a nonlinear ordinary differential equation. The elementary and elliptic function solutions of this equation are investigated. Many exact solutions containing the rational, complex, trigonometric, and hyperbolic functions are also found. Besides, some of the solutions are demonstrated by the graphics. [ABSTRACT FROM AUTHOR]

In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients in two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers.

The derivation of a nonlinear ordinary differential equation for modeling the nonlinear oscillations of a gas bubble placed in an ultrasonic field is performed in terms of bubble-volume variations up to the fourth-order approximation. The equation, written within the Rayleigh-Plesset framework, is solved through numerical approximations. Results from simulations are compared to data obtained from the classic second-order approximation equation derived in the 1960–70’s, usually used in this framework, and from the third-order approximation equation derived in the 1990’s. This comparison shows that the fourth-order approximation allows us to observe the nonlinear behavior of the bubble at high finite amplitude, which differs from the other approximations when the nonlinearity of the phenomenon is higher, i.e., when the driving acoustic frequency is close to the bubble resonance.

Summary: New explicit one‐step high‐order numerical methods (Taylor series and Runge‐Kutta methods) for singular initial value problems for second‐order nonlinear ordinary differential equations are constructed. These methods allow to calculate an approximate solution in the neighborhood of a singular point x=0 with a high‐order of accuracy. By the substitution of the independent variable, we transform the original singular initial value problem into a nonsingular one at an infinite interval. To solve this nonsingular problem, standard numerical methods with initial conditions obtained near a singular point can be used. The effectiveness of the presented approach is confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

The usage of the nanomaterial inside the base fluid for hydrothermal enhancement has augmented in various devices and instruments. Though, several works are done to study the hydrodynamic of the nanofluid in different models and problems. In this current study, analytical approach is used to reveal the influence of nanomaterial on MHD flow in non-parallel plates. Homotopy perturbation method (HPM) is recently applied as reliable techniques for solving the nonlinear governing equations of the engineering problems. This study applied HPM as well as numerical technique of a fourth-order RK4 to ensure the validity of the results. Parametric research is done to find the effect of nanofluid fraction, Hartmann number and geometry angle on velocity profiles. According to our findings, hydraulic boundary layer thickness drops with growth of Re and it rises as Ha augments. [ABSTRACT FROM AUTHOR]

For the second-order nonlinear ordinary differential equation u xx ' ' = u 5 − tu 3 − x , we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation u5− tu3− x = 0. For this solution, we find an asymptotics, which is uniform on the ray t ∈ (−∞,−Mt) as x → +∞; separately, we construct asymptotics on the ray s > Ms and on the segment 0 ≤ s ≤ Ms, where s = |t|−5/2x is the variable compressed with respect to x. Using the method of matching of asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction t ≤ 0 as x2 + t2→∞. [ABSTRACT FROM AUTHOR]

A precise determination of the corneal shape and predicting the changes in the corneal geometry can be of paramount importance for early detection of certain eye diseases and for assessing the postoperative outcomes of refractive surgery. In this article, a simple approach is proposed to solve a nonlinear differential equation that models the human corneal shape. Based on this approach, three approximate analytical solutions are developed using linear solution and Taylor series method. The accuracy and the convergence rate of the solutions are evaluated for different values of constant parameters of the equation that can arise in various physical situations. Our results show that the proposed solutions are more accurate than the other approximate analytical solutions available in the literature. In comparison with the homotopy perturbation method and perturbation method, our approach has less computational complexity.

In hydraulic engineering, the steady non-uniform flow in a channel with the gradual changes at the water surface level is introduced as the Gradually Varied Flow (GVF). For the design of open channels, it is necessary to calculate the GVF profile along the channel flow. The GVF profile is described by a nonlinear Ordinary Differential Equation (ODE). Because this equation is strongly nonlinear, providing new analytical and/or semi-analytical solutions for this equation without any simplifications and/or linearizations would be necessary and helpful. In this research, the Perturbation Method (PM) is proposed to present a semi-analytical solution for solving the GVF equation in the prismatic triangular channel. A total of two cases are studied in this paper. In case 1, the Manning equation and in case 2, the Chezy equation are applied as the resistance equations. The GVF profiles in the two cases are compared with the Finite Difference Method (FDM) profiles. Also, the effect of the summation truncation in the PM is studied for these cases. The results show that by increasing the terms approximation in the PM, the GVF profile converges to the FDM profile. A reference solution for efficiency assessment of numerical techniques can be provided by presented semianalytical solutions in this paper. Furthermore, the proposed method in this paper can be used as a new idea in providing semi-analytical solutions to other open channel works. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of contact transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented. [ABSTRACT FROM AUTHOR]

In this article, we define a p-generalized modified error function as the solution to a non-linear ordinary differential equation of second order, with a Robin type boundary condition at x = 0. We prove existence and uniqueness of a non-negative C∞ solution by using a fixed point argument. We show that the p-generalized modified error function converges to the p-modified error function defined as the solution to a similar problem with a Dirichlet boundary condition. In both problems, for p=1, the generalized modified error function and the modified error function are recovered. In addition, we analyze the existence and uniqueness of solution to a problem with a Neumann boundary condition. [ABSTRACT FROM AUTHOR]

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval (− ∞ , ln ⁡ a ]. On some finite irregular grid { t n ∈ (− ∞ , ln ⁡ a ] , n = 0 , 1 ,... , N , t N = ln ⁡ a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. [ABSTRACT FROM AUTHOR]

Landesman-Lazer’s type efficient sufficient conditions are established for the solvability of the Dirichlet problem , for where ;R) and f is the L([a,b]; R) Caratheodory function, in the case where the linear problem has nontrivial solutions. The results obtained in the paper are optimal in the sense that if , i.e., when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm’s theorem.

Landesman-Lazer's type efficient sufficient conditions are established for the solvability of the Dirichlet problem u"(t) = p(t)u(t) + f(t, u(t)) + h(t), for a ≤ t ≤ b, u(a) = 0, u(b) = 0, where h, p ε L([a, b],R) and f is the L([a, b]; R) Caratheodory function, in the case where the linear problem u'(t) = p(t)u(t), u(a) = 0, u(b) = 0 has nontrivial solutions. The results obtained in the paper are optimal in the sense that if f ⲷ 0, i.e., when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm's theorem. [ABSTRACT FROM AUTHOR]

In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients in two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers. [ABSTRACT FROM AUTHOR]

In the present paper, we have applied the optimal auxiliary function method (OAFM) for obtaining an analytical solution for solving unsteady flow of gas through a porous medium, which is strongly nonlinear boundary value problem on semi-infinite domain. Our technique finds the solution without any perturbation, discretization, linearization or restrictive assumption as it is like the existence of small or large parameters. This procedure reduces the solutions of this nonlinear problem to the solutions of two linear differential equations, avoiding the complexity provided by other known techniques. Finally, by comparing the results obtained through OAFM with some other procedures, it shows that our technique is effective, highly efficient and controls the convergence approximate solutions using a very fast convergence after only one iteration. Also, numerical values are performed in order to validate the analytical results. The results obtained in this paper are very good in comparison with the numerical results and those known in the literature. • The problem of unsteady flow of gas through a porous medium is analytically studied. • Closed-form expression for variable is obtained on large domain, larger than usually. • The most important advantage of our technique is given by the high accuracy of result. • The OAFM proves to be a useful tool to handle complicated nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution