Nonlinear Second-order Partial Differential Equations Research Articles

On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates

The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (α<sub>12</sub>), ratio of thermal conductivity (<i>k</i><sub>12</sub>), dimensionless temperature (θ<sub><i> f</i></sub>), Fourier number (F<sub>0</sub>), Stefan number (Ste), and ratio of densities (ρ<sub>1</sub> / ρ<sub>2</sub>) are discussed in detail.

Mathematical Model and Theoretical Investigation of the Performance of Journal Bearing using a discretized Reynolds Lubrication Equation with Finite Width

Hydrodynamic journal bearings are widely used for a large number of applications such as rotating turbo machinery, axial flow compressors, axial flow turbines, and internal combustion engines. These types of journals are used to support high loads high speed rotating turbo machinery elements mainly shafts and attached components and also for enhancing the quality of these machines. In this work, the importance of journal bearing and an overview of the Reynolds hydrodynamic equation in three dimensions have been studied using the Boundary Value finite-difference method. The second-order nonlinear partial differential equation has been solved using numerical iterations approach with MATLAB software. The numerical solution of the transient Reynolds equation has been studied to investigate the Pressure distribution, pressure gradient as well film lubricant thickness on journal bearings. A numerical solution using ANSYS FLUENT has been applied to investigate the main parameters that have the main influence on the performance of the journal bearing and bearing performance.

Robust [formula omitted] synchronization for parabolic complex networks with coupling time-varying delays

This paper delves into the robust H∞ synchronization for diffusion-type parabolic complex networks (PCNs) characterized by second-order nonlinear partial differential equations (PDEs). Given that the potential deterioration of network performance or the loss of network synchronization caused by external disturbances and time delays, the analysis and synthesis results for the robust H∞ synchronization of the PCNs with coupling time-varying delays are derived with and without the utilization of the distributed adaptive state feedback control protocol. A numerical example is employed to validate the effectiveness of the acquired H∞ synchronization results and the adaptive protocol.

Liouville integrable binomial Hamiltonian system

In this study, we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion of the system. From a mathematical point of view, the equations of motion can be written as integrable second-order nonlinear partial differential equations in 1 + 1 dimensions.

Heat transfer analysis of MHD viscous nanofluid flow over a nonlinearly stretching sheet with heat source/sink: A numerical study

The present study analyses the magnetohydrodynamics (MHD) viscous nanofluid flow past a nonlinearly stretching sheet embedded in a porous medium under the influence of heat generation and thermal radiation. Further, variable magnetic field as well as variable permeability is used instead of constant magnetic field and permeability due to many therapeutic applications, which makes them distinctive and practically helpful. The MHD viscous flow and heat transfer equations have been generated as coupled second-order nonlinear partial differential equations using a mathematical model that resembles the physical flow problem. The partial differential equations governing the flow problems are reduced to ordinary differential equations via similarity variables. The reduced equations are then solved by Runge–Kutta–Fehlberg’s scheme with shooting method by the help of MATLAB software. The result obtained reveals that enhancing the heat generation parameter, porosity parameter, and radiation parameter upsurges the temperature of the nanofluid, whereas the magnetic parameter shows the reverse effect in the case of velocity distribution. Nusselt number decreases at the rate of 2.6147% for copper–water nanofluid when increased from 0 to 0.05. Silver nanoparticles have the highest heat transfer as compared to other nanoparticles. Because of this, silver nanoparticles are the most effective thermal conductors for increasing the effectiveness of heat storage in phase transition materials.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Developments on the Stability of the Non-symmetric Coupling of Finite and Boundary Elements

Abstract We consider the non-symmetric coupling of finite and boundary elements to solve second-order nonlinear partial differential equations defined in unbounded domains. We present a novel condition that ensures that the associated semi-linear form induces a strongly monotone operator, keeping track of the dependence on the linear combination of the interior domain equation with the boundary integral one. We show that an optimal ellipticity condition, relating the nonlinear operator to the contraction constant of the shifted double-layer integral operator, is guaranteed by choosing a particular linear combination. These results generalize those obtained by Of and Steinbach [Is the one-equation coupling of finite and boundary element methods always stable?, ZAMM Z. Angew. Math. Mech. 93 (2013), 6–7, 476–484] and [On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems, Numer. Math. 127 (2014), 3, 567–593], and by Steinbach [A note on the stable one-equation coupling of finite and boundary elements, SIAM J. Numer. Anal. 49 (2011), 4, 1521–1531], where the simple sum of the two coupling equations has been considered. Numerical examples confirm the theoretical results on the sharpness of the presented estimates.

Numerical solution of second-order nonlinear partial differential equations originating from physical phenomena using Hermite based block methods

A Hermite based block method (HBBM) is proposed for the numerical solution of second-order non-linear elliptic partial differential equations (PDEs). The development of the method was accomplished through the methodology of interpolation and collocation procedures. The method’s analysis reveals that it satisfied the requirements for a numerical technique to be convergent. The implementation of the method is extensively discussed. Five numerical examples originating from physical phenomena are presented, and the applicability and accuracy of the HBBM are established by comparing them with the existing methods; the haar wavelet collocation method, the modified cubic B-spline collocation method, and the modified decomposition method. The proposed methods of HBBM are more accurate, stable, and convergent

The radial slump of a gravity current in a confined porous layer

The flow of a gravity current widely occurs in many geophysical, environmental and industrial processes. We focus on the overlooked process of radial slump of a gravity current in confined porous layers, which is related to the post-injection flow dynamics of many subsurface displacement processes, initiated by fluid injection through vertical wells. Specifically, we describe the time evolution of the interface shape and location of the propagating front, governed by a second-order nonlinear partial differential equation (PDE). By solving a nonlinear eigenvalue problem, we identify a self-similar solution to describe the dynamics of confined flows at early times, when the interface attaches to both the top and bottom boundaries of the porous layer. As time progresses, the flow gradually becomes unconfined, as gravity-driven slump causes the interface to detach from one boundary; the interface evolution at late times is then described by a self-similar solution of a nonlinear diffusion equation. Time transition from the early-time confined towards late-time unconfined self-similar solutions is also verified by comparing the self-similar solutions and numerical solutions of the governing PDE for both the interface shape and the frontal location. Interestingly, two special vertical locations have also been identified where the best agreement appears between the PDE numerical solutions and the early-time self-similar solutions for the interface shape, which is a novel feature of the radial flow, and the degree of agreement is not impacted by the time evolution.

Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

In this study, we present the stability analysis of a fully explicit finite difference method (FDM) for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear partial differential equation (PDE), which describes the antiphase boundaries of the binary phase separation. In the presented stability analysis, we consider the explicit Euler method for the temporal derivative and second-order finite difference in the space direction. The explicit scheme is fast and accurate because it uses a small time step, however, it has a temporal step constraint. We analyze and compute that the explicit time step constraint formula guarantees the discrete maximum principle for the numerical solutions of the AC equation. The numerical stability of the explicit scheme automatically holds when we use the time satisfying the discrete maximum principle. The computational numerical experiments demonstrate the stability, discrete maximum principle, and accuracy of the explicit scheme for the constrained time step. Furthermore, it is shown that the time step obtained is not severe restriction when we consider the temporal accuracy.

An optimal control problem without control costs.

A two-dimensional diffusion process is controlled until it enters a given subset of $ \mathbb{R}^2 $. The aim is to find the control that minimizes the expected value of a cost function in which there are no control costs. The optimal control can be expressed in terms of the value function, which gives the smallest value that the expected cost can take. To obtain the value function, one can make use of dynamic programming to find the differential equation it satisfies. This differential equation is a non-linear second-order partial differential equation. We find explicit solutions to this non-linear equation, subject to the appropriate boundary conditions, in important particular cases. The method of similarity solutions is used.

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.

Analysis of oscillations in 1D nanostructures influenced by different concrete potential functions

A family of nonlinear second-order partial differential equations based on concrete potential functions were formulated and analyzed. The occurrence of linear and nonlinear wave solutions were dependent on the interaction potential introduced into the nanostructure and on the initial conditions introduced to the system. The family of nonlinear equations were numerically analyzed because at present, it is difficult (or not possible) to obtain their exact solutions. The law of linear growth was introduced into the nanostructure and space variables were transformed in terms of new variables. The accreting nanostructure was linearly growing for the interaction potentials introduced. Due to the nonlinear form of the formulated equations, the method of lines approach was used to obtain the numerical solutions of the governing equations. Copper atoms were used to in all simulations and calculations.

Fuzzy solution of nonlinear Boussinesq equation

Abstract In this paper, the solution of the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation is examined. The physical problem concerns unsteady flow in a semi-infinite unconfined aquifer bordering a lake. In the examined problem, there is a sudden rise and subsequent stabilization of the lake's water level, thus the aquifer is recharging from the lake. The aquifer boundary conditions are considered fuzzy and, therefore, ambiguities are created to the solution of the overall physical problem. Then, the fuzzy problem is translated to a system of crisp boundary value problems. By using a Boltzmann transformation, the crisp problem is transformed into an integro-differential equation and solved with the help of a special numerical method. This method has a simple iterative procedure, which converges rapidly and is proven very accurate in comparison with other analytical methods. Additionally, the algebraic equation estimates very close to the storage coefficient, the flux at the stream-aquifer origin, and the water stored volume with respect to other exact solutions. This method is very useful for practical cases (artificial recharge problems, irrigation, and drainage projects), giving the possibility to decision-makers to take the right decision.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument.

An Application of Variational Minimization: Quasi-Harmonic Coons Patches

For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation ( p d e ), whose solution in general is the desired surface as the unknown function of surface parameters. The solution of this partial differential equation is known only for very few cases. Instead of solving the corresponding partial differential equation, we exploit an ansatz method (Ahmad et al. (2013), Ahmad et al. (2014) Ahmad et al. (2015)), used for Coons patches spanned by finite number of boundary curves for a quasi-minimal surface as the extremal of r m s of mean curvature. The ansatz method targets a slightly perturbed surface (the rational blending interpolants-based Coons patch, Hermite cubic polynomials interpolants-based Coons patch, and Ferguson surface (vanishing twist vectors) in our case) that comprises initially a nonminimal surface plus the product of a real parameter with a variational function of surface parameters (vanishes at the boundary curves along the unit normal to the slightly perturbed surface). The variational function can be deliberately chosen as the product of linear functions and the mean curvature of the initial nonminimal surface such that it is zero at the boundary curves and then replace this mean curvature by the mean curvature of the resulting surface to find a surface of reduced area, and the process can be repeated for further improvement. In this article, we extend the ansatz method (1) for the rational blending functions interpolants-based Coons patch with a parameter in their form for its different values and (2) for the blending functions comprising Hermite cubic polynomial interpolants-based Coons patch (bicubically blended Coons patch (BBCP) and the Ferguson surface). The ansatz method can be extended for the variational extremal of the surfaces for fuzzy optimal control problems (Filev et al. (1992), Emamizadeh (2005), Farhadinia (2011), Mustafa et al. (2021)).

Optimal social welfare policy within financial and life insurance markets

We consider a continuous lifetime model for investor whose lifetime is a random variable. We assume the investor has an access to the social welfare system, the financial market and the life insurance market. The investor aims to find the optimal strategies that maximize the expected utility obtained from consumption, investing in the financial market, buying life insurance, registering in the social welfare system, the size of his estate in the event of premature death and the size of his fortune at time of retirement if he lives that long. We use dynamic programming techniques to derive a second-order nonlinear partial differential equation whose solution is the maximum objective function. We use special case of discounted constant relative risk aversion utilities to find an explicit solutions for the optimal strategies. Finally, we have shown a numerical solution for the problem under consideration and study some properties for the optimal strategies.