Order Nonlinear PDE Research Articles

Numerical and optimal thermal analysis of non-Newtonian fluids flow over a wedge by using response surface methodology (RSM): Sensitivity analysis

The numerical and optimal thermal analysis of Casson fluid flow over a wedge is investigated in this work. To perform numerical and sensitivity analysis we have convert third order non-linear PDEs into set of ODEs by using similarity transformations along with boundary conditions. These equations are simulated using numerical technique Keller Box method and then compared the values of skin friction coefficient and Nusselt number with those existing in literature and found excellent agreement. Then by using these numerical data we have performed sensitivity analysis by using Response Surface Methodology (RSM). ANOVA tables are generated and developed a correlations between input parameters and output responses. The coefficient of determination for output responses skinfriction coefficient and Nusselt number are 96.50% and 99.51% respectively which shows the best fitted correlations. Finally, we have performed sensitivity analysis which reflects that skinfriction coefficient is more sensitive to Falkner–Skan exponent (m) and there is no effect of Prandtl number (Pr) on skinfriction coefficient. It is also concluded that Nusselt number is sensitive to Prandtl number (Pr) and obtains its optimum value with variation in Prandtl number (Pr).

New exact solutions to space–time fractional telegraph equation with conformable derivative

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph equation plays crucial role in the mathematical study of voltage and current on an electrical transmission line. In an effort to motivate the researchers, we have used very efficacious technique called IGREM method to find a variety of soliton solutions, in which appropriate conformable fractional wave transformation converts nonlinear partial fractional differential equation into nonlinear ordinary equation which is then solved to find the exact solutions of equation. Hence, we succeed in constructing rich solutions that include periodic wave, singular periodic wave solutions, dark, combined dark–bright and rational solutions. To investigate wave profiles of various solutions, 2D and 3D graphs have been explained. The originality of this work has been proved by comparing the obtained results with previously present results in literature. This method has substantial advantage over other methods that it can be used to solve higher order nonlinear PDEs, and this method allows to solve these equations more conveniently with the aid of mathematical software.

On a novel approach for the investigation and approximation of solutions to the systems of higher order nonlinear PDEs

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.

Controlling monotonicity of nonlinear operators

Controlling the monotonicity and growth of Leray–Lions’ operators including the p-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDEs. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.

Numerical study of high order nonlinear dispersive PDEs using different RBF approaches

To realize and comprehend the physical phenomena of nonlinear system, exploration of traveling wave solutions plays an important role. Among the class of dispersive PDEs of traveling wave solutions the Degasperis-Procesi (DP) equation comprises high order nonlinear derivatives and is considered as a well known model for shallow water dynamics having similar asymptotic accuracy as for the Camassa-Holm (CH) equation. In this study we investigate solutions of some high order nonlinear dispersive PDEs namely generalized Degasperis-Procesi (DP), Camassa-Holm (CH) and Korteweg-de Vries (KdV) equations by the use of Radial Basis Function (RBF) combined with Finite Differences (RBF-FD) and Pseudo-Spectral (RBF-PS) methods. For the time derivative approximation, the fourth-order Runge-Kutta (RK) technique is accomplished. The efficiency and accuracy of our suggested approaches are demonstrated using examples and results.

An improved coupled PDE system applied to the inverse image denoising problem

<abstract><p>The problem of interest in this paper is the mathematical and numerical analysis of a new non-variational model based on a high order non-linear PDE system resulting from image denoising. This model is motivated by involving the decomposition approach of $ H^{-1} $ norm suggested by Guo et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>]</sup> which is more appropriate to represent the small details in the textured image. Our model is based on a diffusion tensor that improves the behavior of the Perona-Malik diffusion directions in homogeneous regions and the Weickert model near tiny edges with a high diffusion order. A rigorous analysis of the existence and uniqueness of the weak solution of the proposed reaction-diffusion model is cheked in a suitable functional framework, using the Schauder fixed point theorem. Finally, we carry out a numerical result to show the effectiveness of our model by comparing the results obtained with some competitive models.</p></abstract>

Well-posedness and simulation results of a coupled denoising PDE

In this paper, we are interested in the mathematical and numerical study of a new non variational model based on a high order non-linear PDE system for image denoising. The proposed model generalizes and improves the nonvariational model of Halim and Kumar (2020). Our model is based on a diffusion tensor that improves the behavior of the Perona–Malik diffusion directions in homogeneous regions and the Weickert model near tiny edges with a high diffusion order. The proposed approach takes into consideration the best of regularized Perona–Malik process in flat regions, and the benefit of the Weickert filter effect near sharp edges. To overcome the difficulty of the 4th order in the proposed equation, we involve the decomposition approach of H−1 norm suggested by Guo et al. (2011) [20, 21] to transform it into coupled second order equations which is more appropriate to represent the small details in the textured image. By the Schauder method, we establish the existence and uniqueness of the weak solution of the proposed reaction–diffusion System. In the end, some experiment results illustrate the effectiveness of the proposed model compared with the results of some existing models.

A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements

In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical PDE, namely the Cahn–Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry—and mesh—characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments.

Efficient Method for Solving Fourth Order PDEs

The aim of the article is to implement the LA Transform with HPM to solve the 4th order non-linear PDEs arising in mathematical physics and astrophysics. This method is based on a combination of LA Transformation with HPM without any discretization or restrictive assumptions. Some well-known 4th order non-linear PDEs are solved as illustrative examples to demonstrate the accuracy, efficiency and easy implementation of this technique.

Strong Solutions for PDE-Based Tomography by Unsupervised Learning

We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We employ an unsupervised approach such that the input to the network is a point set in an arbitrary domain, and the output is the set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit, smooth, differentiable function with a known analytical form. We solve the forward problem (observations given the underlying model's parameters), semi-inverse problem (model's parameters given the observations in the whole domain), and full tomography inverse problem (model's parameters given the observations on the boundary) by solving the forward and semi-inverse problems at the same time. The optimized loss function consists of few elements: fidelity term of $L_2$ norm that enforces the PDE in the weak sense, an $L_\infty$ norm term that enforces pointwise fidelity and thus promotes a strong solution, and boundary and initial conditions constraints. It further accommodates regularizers for the solution and/or the model's parameters of the differential operator. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape two dimensional (2D) second order systems with application to electrical impedance tomography (EIT) and diffusion equation. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework enables, in principle, the solution of high order and high dimensional nonlinear PDEs.

Characterizing envelopes of moving rotational cones and applications in CNC machining

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.

A [formula omitted] PDE model for effective denoising

In this paper, we have proposed a higher order non-linear PDE model containing diffusion terms of TV−L2 and TV−H−1 model for effective image denoising. Fourier spectral method for space and convexity splitting method for time have been used to solve the proposed PDE model. Stability analysis for the time discretized equation has been derived. Numerical experiment has been done on some test images and the results have been compared with the results of some existing models.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for d=1 and the beam equation for dle 3. We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.

Numerical modeling of phase separation on dynamic surfaces

The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn–Hilliard equation posed on a time-dependent surface. Only elementary tangential calculus and the embedding of the surface in R3 are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. A hybrid method, finite difference in time and trace finite element in space, is introduced and stability of its semi-discrete version is proved. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the equation. Thus, the method is capable of solving the Cahn–Hilliard equation numerically on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. We assess the approach on a set of test problems and apply it to model spinodal decomposition and pattern formation on colliding surfaces. Finally, we consider the phase separation on a sphere splitting into two droplets.

On a Critical Fourth Order PDE with Navier Boundary Condition

We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of ℝn, n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.

Numerical study for fractional model of non-linear predator-prey biological population dynamical system

The key objective of the present paper is to propose a numerical scheme based on the homotopy analysis transform technique to analyze a time-fractional non-linear predator-prey population model. The population model are coupled fractional order non-linear PDE often employed to narrate the dynamics of biological systems in which two species interact, first is a predator and the second is a prey. The proposed scheme provides the series solution with a great freedom and flexibility by choosing appropriate parameters. The convergence of the results is free from small or large parameters. Three examples are discussed to demonstrate the correctness and efficiency of the used computational approach.

Unsteady MHD blood flow through porous medium in a parallel plate channel

In this study, we have analyzed heat and mass transfer effects on unsteady blood flow through parallel plate channel in a saturated porous medium in the presence of a transverse magnetic field with thermal radiation. The governing higher order nonlinear PDE’S are converted to dimensionless equations using dimensionless variables. The dimensionless equations are then solved analytically using boundary conditions by choosing the axial flow transport and the fields of concentration and temperature apart from the normal velocity as a function of y and t. The effects of different pertinent parameters appeared in this model viz thermal radiation, Prandtl number, Heat source parameter, Hartmann number, Permeability parameter, Decay parameter on axial flow transport and the normal velocity are analyzed in detail.

Formula omitted]-discontinuous Galerkin methods for a wind-driven ocean circulation model: Two-grid algorithm

This paper presents a nonconforming finite element method for a streamfunction formulation of the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. The streamfunction formulation is a fourth order nonlinear PDE and the nonconforming method is based on C0-elements instead of C1-elements. Existence and uniqueness of the approximation are proved and optimal error estimates in several norms of interest are demonstrated under a small data assumption. Two-grid algorithms based on Picard and Newton type linearizations are then presented to efficiently resolve nonlinearities and computational results are given to demonstrate the efficiency of the algorithm. The Mediterranean sea example is tested with real world coastline data, which illustrates the effectiveness of the two-grid approach in the wind-driven ocean circulation simulation.

On Microlocal Smoothness of Solutions of First Order Nonlinear PDE

On Microlocal Smoothness of Solutions of First Order Nonlinear PDE

Two-layer spin coating flow of Newtonian liquids: A computational study

Spin coating is the most commonly used method in industries to form coating films of desired thickness and functionality. In case of two-layer spin coating process, stratified layers of two immiscible liquids are deposited onto the substrate simultaneously, which spread and thin to form two-layer coating film of finite thickness. Questions concerning the effect of surface/interfacial tension on uniformity of these films and contact line evolution are relevant and need thorough investigation. Therefore, an axisymmetric model governing the flow of two-layer spin coating process is developed here. Liquids used for the study are assumed to be Newtonian and fully wetting. The contact line singularity is resolved using a precursor layer model and the governing equations are simplified using lubrication approximation. A Galerkin finite-element method (G/FEM) based scheme is developed to solve the resulting fourth order non-linear PDEs. Simulation results reveal that the fluid properties like ratio of the viscosity of upper layer fluid to lower layer and ratio of the upper gas-liquid surface tension to inner liquid-liquid interfacial tension have profound impact on the time evolution of the film profile, contact radius and shape of the capillary ridges. It is observed that a uniform two-layer film surrounded by thin single layer film is formed when the viscosity ratio is small. On the contrary, when viscosity ratio is large, a thin two-layer film surrounded by bulky capillary ridges is formed. Similarly, the results also show that sharpness of capillary ridge increases with decrease in the surface tension ratio. Further, it is found that increase in the precursor layer thickness increases the spreading rate, thereby making the film more uniform. Finally, the uniformity of the final two-layer film does not get affected by the initial volume of fluid present in the upper layer.