Partial Differential Equations in Applied Mathematics最新文献

In this study, we devise a parameter-uniform second-order numerical method for two-parameter singularly perturbed partial differential equations with large time lag. The equations are discretized using the Crank–Nicolson method in time direction on uniform mesh and the cubic spline method in space direction on a Bakhvalov mesh. The theoretical parameter-uniform convergence analysis and the numerical results proves that the present method gives second-order $\varepsilon -$uniform convergence both in space and time directions. Two numerical experiments are performed.

Owing to enhanced features of nanomaterials, various applications of such materials have been suggested in the thermal systems, heat exchanges, electronic cooling systems, coolant processes, energy production etc. Following to such motivated applications, the objective of current analysis is to presents the bioconvective double stratification flow of Sutterby nanofluid due to bidirectional moving surface. The thermal interpretation of problem is subject to utilization of radiative effects, activation energy and heat source. The observations for heat, mass and microorganism's assessment are performed by using the convective boundary conditions. Shooting numerical simulations are performed for modeled problem. It is noticed that heat transfer enhances due to thermal stratification parameter. An increment in mass transfer is subject to larger values of mass stratification Biot number. The claimed results offer significance in controlling the heating and cooling processes, thermal devices, energy generation, manufacturing developments, solar systems etc.

The nanofluids are decomposition of nano-sized metallic particles with base materials maintaining peak thermal properties. Owing to enhanced thermal features, various applications of nanofluids are observed in enhancing energy resources and cooling processes. The objective of current work is exploring the thermal impact of nanofluid associated to the oblique stagnation point flow. The thermal interpretation of nanofluid is subject to active and passive control approach. The heat trnasfer analysis is identified by using convective thermal flow constraints. The Buongiorno nanofluid model is adopted, endorsing the Brownian and thermophoresis consequences. The flow problem is first simplfied intodimensionless form. The numerical computations are performed by using famous shooting scheme with justifiable accuracy. A comparative change in the thermal phenomenon given the passive and active frameworks is presented. It is exmained that heat transfer reduces for stretching ratio parameter. The concentration profile reduces for angle of incidence for both active and passive control approach.

A cohomology theory for “odd polygon” relations—algebraic imitations of Pachner moves in dimensions 3, 5, …—is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example calculation results are presented.

In the current research work, the classical wave equation is combined with a nonlinear $sinh$ source term in the $sinh$-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

This paper presents a Lie group analysis of the Schamel Burger’s equation, notable for producing shock-type traveling waves in distinctive physical contexts. We determine the infinitesimal generators for this equation using the Lie group theory of differential equations. By applying Lie point symmetries, we establish commutation relations, the adjoint representation, and identify the optimal system of sub-algebras. Using elements from this optimal system, we perform symmetry reductions, resulting in various nonlinear ordinary differential equations (ODEs). Some of these reductions yield exact explicit solutions, while others necessitate the use of the new auxiliary equation method to obtain optical soliton solutions. We illustrate the dynamics of these soliton solutions graphically through both two and three-dimensional representations of wave structures. Additionally, we compute the conservation laws for the Schamel Burger’s equation by applying Ibragimov’s theorem, deriving conserved quantities corresponding to its point Lie symmetries. This analysis underscores our novel contribution, offering insights not previously explored in the literature.

This paper examines a one-dimensional (1D) model that appears in arterial blood flow. The mathematical model for blood flow via arteries is similar to that of unstable incompressible flows in thin-walled collapsible tubes. We present the Riemann invariants of the suggested model, which is one of the fundamental components of this work. For numerical modeling of blood flow model, we present a nonhomogeneous Riemann solver (NHRS) technique. Next, we demonstrate the simulation of how pressure, velocity, and cross section area waveforms propagate through arteries. Specifically, we present numerical test cases with various initial data sets. In addition, we compare the NHRS scheme to the classic Rusanov, Lax–Friedrichs, and Roe schemes. Theoretical models for thin-walled collapsible tubes are applicable to a wide range of physiological events and may be used to build clinical devices for actual biomedical science. The NHRS method’s accuracy and efficiency are demonstrated by the numerical tests.

The mathematical modeling and dynamic analysis of time-delayed enzymatic chemical reactions in biological systems are presented in this research. The objective is to examine the function of time lags in these reactions and to get a complete knowledge of the behavior of biological systems in a reaction to modifications in the quantity present of reactants and products. The model, which is based on delay differential equations, includes a time delay term to account for the lag between changes in the concentration of reactants, reaction rate constants and product responses. The findings give insight into how enzymatic processes behave dynamically and how stability is impacted by time lags, oscillation and general efficiency of the system. These results have significant importance for our comprehension of how biological processes are regulated and for the creation of biological control structures.

In this paper, studied the impact of heat generation of Nanofluid movement over a stretching sheet by the consideration of Thermophoresis, Brownian motion & first order chemical react parameters etc. Constructed the modelling equations with based on assumptions and by introducing emerging parameters. The equations converted to third order ODE through stream functions. FDM with collocation polynomial technique (bvp4c) employed to solve those equations through MATLAB software. The results are presented through graphical form with the influence of emerging parameters. Thickness of thermal boundary stratum decreased as enhancing of Prandtl number. Influence of Brownian motion parameter, fluid temperature raised and fall down the concentration. Temperature of fluid and concentration raised as enhancement of thermophoresis. A decrease in the heat transfer rate and an increase in the mass transfer rate are observed as thermophoresis, Brownian motion, and heat generation parameter values increasing. The enhancement of chemical reaction parameters intensifies the driving forces of temperature and concentration gradients, which govern heat and mass transfer, leading to increased rates of both heat and mass transfer. Validation of the model presented and the present results align well by past reported studies. This model can extent to analyse the hybrid nanofluid in the manufacturing process of detergent, painting and lubricants, analysis of blood flow in artery etc.

In this manuscript, the perturbed nonlinear Schrödinger’s equation (PNLSE) is considered, which has many implications in various fields such as ferromagnetic material, nonlinear optics, and optical fibers. The focus of this paper is to obtain soliton solutions of the perturbed nonlinear Schrödinger’s equation by implementing two analytical methods, namely, tanh–coth method and energy balance method. As an outcome, a various soliton solutions like, breather solitary wave, lump type soliton in periodic background, singular type soliton, and periodic soliton solution obtain via Mathematica. Additionally, the 2D, 3D, and contour plots are used to visualize the graphical propagation of the achieved soliton solutions by selecting appropriate parametric values.