Solutions Of Differential Equations Research Articles

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

A Note On Lie Symmetry Analysis, Optimal System,New Solitary Wave Solutions And Conservation Laws Of The Pavlov Equation

In this essay, based on the Lie group analysis method, the investigation has been carried out on the symmetry properties of the (2 + 1)-dimensional Pavlov equation. Its maximal symmetry group in Lie's sense and the corresponding one-dimensional optimal system of subgroups are presented. Furthermore, by using Ibragimov's method which is a generalization of Noether's theorem, the conservation laws for the intended equation are determined. Finally, utilizing the traveling wave solutions method, some exact solutions of this nonlinear partial differential equation are constructed.

Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability

This paper deals with the soliton solutions for beam movement within a multimode optical fiber featuring a parabolic index shape. It is considered that a Two-Dimensional Nonlinear Schrödinger Equation (2D-NLSE) with an instantaneous Kerr nonlinearity of the kind can represent the beam dynamics. Nonlinear Multimode Optical Fibers (MMFs) of this kind are gaining popularity because they provide novel approaches to control the spectral, temporal, and spatial characteristics of ultrashort light pulses. We gain the optical soliton solutions for the nonlinear evolution beam dynamics using the Jacobi Elliptic Function Expansion (JEFE) method. The exact analytical solution of Nonlinear Partial Differential Equations (NLPDEs) can be achieved with wide application using the effective JEFE approach. These solutions are obtained in the form of dark, bright, combined dark–bright, complex combo, periodic, and plane wave solutions. Additional solutions for Jacobi elliptic functions, encompassing both single and dual function solutions, have been acquired. This approach is based on Jacobi elliptic functions, which will provide us the exact soliton solutions to nonlinear problems. Additionally, we will analyze the Modulation Instability (MI) for the underlying model. Moreover, we show the physical behavior of the beam propagation in a multimode optical fiber the three-dimensional, two-dimensional, and their corresponding contour plots are dispatched using the different values of parameters.

On the Oscillatory Behavior of Solutions of Second-Order Non-Linear Differential Equations with Damping Term

In this paper, we discuss the oscillatory behavior of solutions of two general classes of nonlinear second-order differential equations. New criteria are obtained using Riccati transformations and the integral averaging techniques. The obtained results improve and generalize some recent criteria in the literature. Moreover, a traditional condition is relaxed. Three examples are given to justify the results.

Parametric factorization of non linear second order differential equations

In this paper the factorization method introduced by Rosu & Cornejo-Pérez for second order non linear differential equations is generalized by adding a parameter in order to obtain the general solutions for the mixed quadratic and linear Liénard type equation. The new parametric factorization is used to obtain complete analytic solutions for nonlinear second order differential equations. The parametric factorization introduced in this article reduces to the standard factorization scheme when the parameter goes to zero. As an example, we apply the parametric factorization approach to solve the generalized Fisher equation and the Israel-Stewart cosmological model. The parametric factorization presented in this paper can be used in other non linear mixed Liénard type equations.

Hall current effect on molecules motion in viscous non-Newtonian power law fluid with Joule heating and fluxing model of Cattaneo-Christov

This study examines the molecules motion influences and thermal attributes in power-law non-Newtonian dissipative fluids' magnetohydrodynamic behavior, Joule heating, electromigration, and Cattaneo-Christov effects. Effects of Hall on both flow velocity and temperature, as well as the heat transport are considered. The investigation looks at heat transference that occurs when there is a slippage consequence, how the temperature influences the viscosity of the fluid, and how the potential parameters influence the exponential non-Newtonian flow of fluid via an extensible sheet. Runge-Kutta-Fehlberg's approach, which is based on the shooting procedure, was used to develop computational solutions for nonlinear ordinary differential equations. When exposed to a deformation that is generated by expansion, pressure, or agitation, a dilatant fluid is characterized by a phenomenon in which its viscosity rises and may harden. This phenomenon is referred to as the dilatation phenomenon. High shearing rates provide the impression that it is stable. When expanded, subjected to pressure, or agitated, dilatant fluids become more viscous and transform into solids. Both the non-equilibrium that occurs within the fluid and the reduction of the irreversibility processes that occur within it are caused by the acquired charges that are a result of Hall effect of fluid molecules.

A numerical method for ergodic optimal control of switching diffusions with reflection

The ergodic or long run average cost control problem for diffusions is one of the classical problems of stochastic control that still eludes a completely satisfactory treatment. This is certainly true for the setting in which the systems to be controlled is modeled by the solution of a switching stochastic differential equation with reflection (SSDER). In this paper, we advance in this rather difficult problem by setting forth preliminary results for the unidimensional case. Besides carving out a numerical method, our treatment of the ergodic control in this scenario straddles issues of existence and uniqueness of solution of the SSDER and a verification theorem for the associated HJB equation. We conclude by illustrating the effectiveness of the method considering the control of energy consumption in a large parallel processing computer system composed of one queue and several processing stations.

On some super-congruences for the coefficients of analytic solutions of certain differential equations

On some super-congruences for the coefficients of analytic solutions of certain differential equations

Analytical investigation of space–time shifted nonlocal stochastic Sasa–Satsuma equation using the enhanced modified extended tanh-expansion method for stochastic solitary waves Solutions

In this paper, discover exact solutions of the space–time shifted nonlocal stochastic Sasa–Satsuma equations using the enhanced modified extended tanh-expansion method. This method is a sophisticated mathematical approach for analytical and approximate solution of a nonlinear partial differential equations. The space–time shifted nonlocal stochastic Sasa–Satsuma equations define the behavior of two complex-valued functions, u and v, in a nonlinear system. These equations are complicated to solve analytically because they contain a number of components that describe space–time shifting, nonlocal interactions, and stochastic problems. To increase reliability and effectiveness of the solutions, this method couples the extended tanh-function method with an enhanced method. In order to demonstrate how white noise affects the solutions to the space–time shifted nonlocal stochastic Sasa–Satsuma equations, we also use Matlab to create 3D surfaces and contour graphs of exact solutions.

Unraveling Drug Delivery from Cyclodextrin Polymer-Coated Breast Implants: Integrating a Unidirectional Diffusion Mathematical Model with COMSOL Simulations.

Breast cancer ranks among the most commonly diagnosed cancers worldwide and bears the highest mortality rate. As an integral component of cancer treatment, mastectomy entails the complete removal of the affected breast. Typically, breast reconstruction, involving the use of silicone implants (augmentation mammaplasty), is employed to address the aftermath of mastectomy. To mitigate postoperative risks associated with mammaplasty, such as capsular contracture or bacterial infections, the functionalization of breast implants with coatings of cyclodextrin polymers as drug delivery systems represents an excellent alternative. In this context, our work focuses on the application of a mathematical model for simulating drug release from breast implants coated with cyclodextrin polymers. The proposed model considers a unidirectional diffusion process following Fick's second law, which was solved using the orthogonal collocation method, a numerical technique employed to approximate solutions for ordinary and partial differential equations. We conducted simulations to obtain release profiles for three therapeutic molecules: pirfenidone, used for preventing capsular contracture; rose Bengal, an anticancer agent; and the antimicrobial peptide KR-12. Furthermore, we calculated the diffusion profiles of these drugs through the cyclodextrin polymers, determining parameters related to diffusivity, solute solid-liquid partition coefficients, and the Sherwood number. Finally, integrating these parameters in COMSOL multiphysics simulations, the unidirectional diffusion mathematical model was validated.

Existence of common fuzzy fixed points via fuzzy F-contractions in b-metric spaces

The main goal of this study is to establish common fuzzy fixed points in the context of complete b-metric spaces for a pair of fuzzy mappings that satisfy F-contractions. To strengthen the validity of the derived results, non-trivial examples are provided to substantiate the conclusions. Moreover, prior discoveries have been drawn as logical extensions from pertinent literature. Our findings are further reinforced and integrated by the numerous implications that this technique has in the literature. Using fixed point techniques to approximate the solutions of differential and integral equations is very useful. Specifically, in order to enhance the validity of our findings, the existence result of the system of non-linear Fredholm integral equations of second-kind is incorporated as an application.

Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay

In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results.

Theoretical estimation of the thermal damages of living tissues caused by laser irradiation in tumor thermal therapy

This article aims to provide theoretical predictions for the thermal reactions of human tissues during tumor thermotherapy when exposed to laser irradiation and an external heat source. For the construction of a theoretical study of bioheat transfer, the selection of a suitable thermal model capable of accurately predicting the required thermal responses is essential. The effect of heat production by heat treatment on a spherical multilayer tumor tissue is evaluated using this approach. Analytical solution for the non-homogenous differential equations is derived in the Laplace domain. The study examines the impact of thermal relaxation time on tissue temperature and the subsequent thermal damage. The numerical findings of thermal damage and temperatures are depicted in a graphical representation. This model explains laser treatment, physical events, metabolic support, and blood perfusion. The numerical outcomes of the recommended model are validated by comparing them to the literatures.

On some fixed point results using [formula omitted] simulation functions via w-distance with applications

Based on the notion of class of CG simulation functions, we launch the concept of (ρ,ϑ)-ZG-contraction on the setting of w-distance mappings. We employ our new concept to formulate and prove several fixed point results over the complete metric spaces. Also, we provide some concrete examples to show the usability of the obtained results. Furthermore, we study the existence of a unique solution of nonlinear fractional differential equations, nonlinear integral equations and spring-mass system problems. The results presented in this manuscript generalize many of the results known in the literature.

Solutions of fuzzy fractional differential equations

Solutions of fuzzy fractional differential equations

On the Asymptotics of Solutions of Stochastic Differential Equations with Jumps

On the Asymptotics of Solutions of Stochastic Differential Equations with Jumps

Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples.

Exact and Approximate Solution of Multi-Higher Order Fractional Differential Equations Via Sawi Transform and Sequential Approximation Method

Exact and Approximate Solution of Multi-Higher Order Fractional Differential Equations Via Sawi Transform and Sequential Approximation Method

Approximate Solution of Linear and Nonlinear Partial Differential Equations Using Picard’s Iterative Method

This work discusses a new description of Picard’s iterative method (PIM) to get the approximate or sometimes exact solution of (linear and nonlinear) partial differential equations ,Provided that by examples to illustrate the reliability and ease of this method.

A Solution of Fractal Dynamic Pharmacokinetics Problem

There are numerous uses for fractional differential equations in engineering, physics, and technology. A path for solving the fractal differential equations was examined, and its homogeneous form of was introduced. The notions of Riemann-Liouville fractional derivatives served as the foundation for the journey.. The solutions of the linear non-homogeneous fractal differential equations are given in detail.