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In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.

This research paper attempts to describe the transmission dynamic of zoonotic visceral leishmaniasis with the aid of a mathematical model by considering the asymptomatic stages in humans and animals. The disease is endemic in several countries. Data used in the research are obtained from the literature while some are assumed based on the disease dynamic. The consideration of both asymptomatic and the symptomatic infected individuals is incorporated in both humans and animals (reservoir), as well as lines of treatment for the human population. It is found that the model has two fixed points; the VL-free fixed point and the VL-endemic fixed point. Stability analysis of the fixed points shows that the VL-free fixed point is globally asymptotically stable whenever the basic reproduction number is less than one and the VL-endemic fixed point is globally asymptotically stable whenever the basic reproduction number is greater than one. Sensitivity analysis is conducted for the parameters in the basic reproduction number, and the profile of each state variable is also depicted using the data obtained from the literature and those assumed. The transmission probability from infected sandflies to animals, transmission probability from infected animals to sandflies, per capita biting rate of sandflies of animals, and rate of transfer from symptomatic infected animals to the recovered class are among the most sensitive parameters that have the greatest influence on the basic reproduction number. Moreover, the value of the basic reproduction number is obtained to be 0.98951, which may require further study, as the margin between potential disease control and outbreak is thin.

In this work, we find the results for the folded part projection of the butterfly catastrophe model onto the control space, using methods from catastrophe theory to obtain stability and the catastrophic behavior of finite periodic solutions for some non-linear differential equations. Finally, we have shown that a saddle-node bifurcation, which can be classified as a butterfly mutation, accompanies butterfly surface folding.

This paper develops the combined effects of free convection magnetohydrodynamic (MHD) flow past a vertical plate embedded in a porous medium. The dimensionless coupled non-linear equations are solved to get the approximate analytical expression for the concentration by using the homotopy perturbation method. For all possible values of parameters, skin lubrication, Nusselt number and Sherwood number are derived.

In this study, a stochastic differential equation capable of describing (using the motion function) the automatic manufacturing process of a lithium battery with a sleeve shell is introduced. The boundary-condition modeling method for this type of motion is an ordinary differential equation. The nonlinear equation is found using a dynamic method. The equations of the motions for the assembly process are derived by reducing the order of terms and separating the variables. Both the derivatives and integrals of the motion functions are derived and applied to describe the assembly process for a lithium battery. By using a characteristic constitutive assembly model, a new method to calculate the nonlinear terms is proposed. The main advantage of this method is that it can simplify the problem, which is done by finding the algebraic error in the region space of the motion function. The reliability and the applicability of the method were confirmed using a real-life example. Finally, by solving and improving the Euler model, the fourth-order Kutta method and Taylor series are applied to verify the correctness of the algorithm, and the convergence order of the function and the optimization of the assembly model are discussed. Through a one-dimensional search of the convergence domain, a method is found to make the analytical solution approximate the exact solution and reduce the calculation cost. This method in dynamics and mathematical engineering.

A transverse harmonic seismic wave, which propagates through the upper fractured layer of the Earth towards the Earth's surface, is considered. It is shown that the attenuation of a seismic wave in the Earth's upper crust is due to internal friction, i.e., friction between the sides of micro-fractures. Such a damping mechanism does not work in the deeper layers of the Earth where high lithostatic pressure prevents the movement along fractures. The governing equation for a brittle fractured medium is nonlinear. In the study of wave propagation, an approximate method of harmonic linearization is used. It is shown that internal friction in the upper crust leads to the transformation of a transverse harmonic wave into a shock wave.

The purpose of this study is to describe the outbreak of a novel coronavirus (COVID-19) and review the computational model, which was based on non-linear ordinary differential equations. The main idea developed was to describe the formulation, integration, and application of mathematical models by using data from China, Austria, Poland, and France. The Logistic Model and its solution Sigmoid Curve, and the SIR Model were used for the determination. After taking the measured data, the total number of cases was predicted, and it was found that the results were obtained in good approximation to the measured results. To predict the number of deaths over time, generalized models were developed in response to COVID-19.

The study of Von Kármán swirling flow is a subject of active interest due to its applications in a wide range of fields, including biofuel manufacturing, rotating heat exchangers, rotating disc reactors, liquid metal pumping engines, food processing, electric power generating systems, designs of multi-pore distributors, and many others. This paper focusses on investigating Von Kármán swirling flows of viscous incompressible fluid due to a rotating disk electrode. The model is based on a system of four coupled second-order non-linear differential equations. The purpose of the present communication is to derive analytical expressions of velocity components by solving the non-linear equations using the homotopy analysis method. Combined effects of the slip λ and porosity γ parameters are studied in detail. If either parameter is increased, all velocity components are reduced, as both have the same effect on the mean velocity profiles. The porosity parameter γ increases the moment coefficient at the disk surface, which monotonically decreases with the slip parameter λ. The analytical results are also compared with numerical solutions, which are in satisfactory agreement. Furthermore, the effects of porosity and slip parameters on velocity profiles are discussed.

In this research, a soft computing approach based on a Nature-inspired technique, the Fractional-Order Darwinian Particle Swarm Optimization (FO-DPSO) algorithm, is hybridized with feed-forward artificial neural network (FF-ANN) to suggest and calculate better solutions for non-linear second-order ordinary differential equation (ODE) representing the corneal shape model (CSM). The unknown weights involved in approximate solutions obtained through ANN are tuned with the help of FO-DPSO. To test the robustness of our approach and conditionality of CSM, we have considered several cases of CSM with different aspects of the problem. Solutions obtained by Adam's method are used as a reference point for the sake of comparison. We establish it that FO-DPSO is a suitable technique for tuning the unknown weights involved in the solution designed with ANNs. Our results suggest that the proposed approach is a suitable candidate for solving real-world problems involving differential equations.

The existence of man is dependent on nature, and this existence can be disturbed by either man-made devastations or by natural disasters. As a universal phenomenon in nature, symmetry has attracted the attention of scholars. The study of symmetry provides insights into physics, chemistry, biology, and mathematics. One of the most important characteristics in the expressive assessment and development of computational design techniques is symmetry. Yet, mathematical models are an important method of studying real-world systems. The symmetry reflected by such a mathematical model reveals the inherent symmetry of real-world systems. This study focuses on the contagious model of pine wilt disease and symmetry, employing the q-HATM (q-Homotopy Analysis Transform Method) to the leading fractional operator Atangana–Baleanu (AB) to arrive at better understanding. The outgrowths are exhibited in the forms of figures and tables. Finally, the paper helps to analyze the practical theory, assisting the prediction of its manner that corresponds to the guidelines when contemplating the replica.

We consider solutions, with infinity of zeros, of real second-order linear differential equations. The graph of a solution locates the zeros with a certain precision, which means that zeros are between some successive numerical values of the solution. Once the first zero is calculated, the given equation is converted into a new first-order equation. The zeros of this new equation form a subset of numerical values of the solution of the first order equation.

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

Consider an origin which is fixed at the Earth’s surface, and a rectangular celestial coordinates axes with this origin. The gravity acceleration is the approximate sum of the centrifugal acceleration of the origin relatively to the Earth’s center and the Newton’s gravitational acceleration due to the Earth’s attraction. Thus the selected reference frame is formally an inertial frame. The origin is also a point of a rigid heavy body. The dynamical differential system of absolute motion of the three degrees-of- freedom body is given. Some notable solutions of this system are calculated relatively to a moving laboratory frame with the angular velocity of the Earth.

In the present paper we deal with nonlinear dynamics of microtubules. The structure and role of microtubules in cells are explained as well as one of models explaining their dynamics. Solutions of the crucial nonlinear differential equation depend on used mathematical methods. Two commonly used procedures, continuum and semi-discrete approximations, are explained. These solutions are solitary waves usually called as kink solitons, breathers and bell-type solitons. [Project of the Serbian Ministry of Education, Science and Technological Development, Grant no. III45010]

In the present paper, we have studied three fundamental flows namely Couette, Poiseuille and generalized Couette flows of an incompressible Casson fluid between parallel plates using slip boundary conditions. The equations governing the flow of Casson fluid are non-linear in nature. Analytical solutions of the non-linear governing equations with non-linear boundary conditions are obtained for each case. The effect of the various parameters on the velocity and volume flow rate for each problem is studied and the results are presented through graphs. It is observed that, the presence of Casson number decreases the velocity and volume flow rate of the fluid. Increasing of slip parameter increases the velocity and volume flow rate in both Poiseuille and generalized Couette flows.

The article offers a “conceptual carcass” of the enterprise model, which is based on known results of studies in sociology, biology, system theory and mathematics. The article lists main features of growth of main indicators of economic activity and development of an enterprise. Dynamics of changes at an enterprise is described with a system of non-linear differential equations. One of the global and dominating factors in it is entrepreneurship of personnel, which is an integral part of its labour mentality or organisational culture. The article proves for the first time ever, using mathematical modelling, that namely entrepreneurship, innovation capability, is a comprehensive and dominant factor of enterprise growth and development.

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold. The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually “jumping” through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.