Non-homogeneous Partial Differential Equations Research Articles

Some Exact Green Function Solutions for Non-Linear Classical Field Theories

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

Gyrator potential operator and Lp -Sobolev spaces involving Gyrator transform

This paper presents the properties of Gyrator potential operator G μ l and related L p -Sobolev spaces associated with Gyrator transform (GT). The Schwartz-type space S Δ ( R 2 ) is introduced. Various properties of the kernel of GT is studied including its continuity. Pseudo-differential operators A α , q associated with GT is defined and derived it's continuity on S Δ ( R 2 ) . The Gyrator potential operator G μ l is defined as a pseudo-differential operator associated with a precise symbol and obtained certain fruitful properties. The operator G μ l is extended to a space of distributions. Another integral representation of A α , q is obtained and its L p ( R 2 ) -boundedness result is derived. The spaces H α m , p ( R 2 ) and H Δ m , p ( R 2 ) are defined. It is shown that the operator G μ l is an isometry of H Δ m , p ( R 2 ) . An L p -boundedness result for the operator G μ l is proved. We conclude the article by applying some of the results to show that the analytical solution of certain non-homogeneous partial differential equations belong to these spaces.

Influence of Temperature on The Concentration and thus on the MHD Oscillatory Flow of a Tangent Hyperbolic Fluid Through a Regular Permeable Channel

The aim of this research is to study the effect of temperature and concentration on oscillatory flow under the influence of magneto hydrodynamic (MHD) of a non-Newtonian hyperbolic fluid in a horizontal permeable channel. The problem formulation is non-linear and non-homogeneous partial differential equations. Using the perturbation technique, we solve the momentum equation under the assumptions of long wavelength and very low Reynolds number. After obtaining the solution, we used the “Mathematica 13” program to analyze the results through graphs, and the oscillatory flow and magnetic field were studied on the movement of the fluid, its temperature, and concentration.

Analyze a temperature and MHD peristaltic flow of sutterby fluid through a porous wavechannel in a rotating frame

The aim of this paper is to analyze the temperature and magnetohydrodynamic (MHD) peristaltic flow of non-Newtonian Sutterby fluid through a porous wavechannel under the influence of rotation. The problem formula is non-linear and non-homogeneous partial differential equations. By using the perturbation technique, we solved the momentum equation under the assumptions of a long wavelength and an extremely low Reynolds number. After getting the solution, we have used the "Mathematica 13" program to analyze the results through graphs, and it have been studied the effect of the rotation, magnetic field, viscosity of the fluid and average radius of the tube on the fluid movement and its temperature.

Theoretical Analysis on Thermo-Mechanical Bending Behavior of Timber–Concrete Composite Beams

In this study, an analytical approach is introduced for predicting the bending behavior of a timber–concrete composite (TCC) beam subjected to a mechanical load and a non-uniform temperature field, in which the orthotropy of timber as well as interfacial slip are taken into consideration. The analytical model addresses the non-uniform temperature field using Fourier series expansion based on the heat transport theory. The stresses and displacements of the TCC beam under the thermo-mechanical condition are governed by the thermo-elasticity theory, and the corresponding solution is derived analytically by solving a group of non-homogeneous partial differential equations. The proposed solution is in good agreement with the finite element solution and exhibits higher accuracy compared to the Euler–Bernoulli beam solution that relies on the assumption of transverse shear deformation and isotropy. An extensive investigation is carried out to analyze how the bending behavior of TCC beams is influenced by variations in interfacial shear stiffness and temperature field.

Thermomechanical modeling of dissimilar-material interfaces in composite structures

Due to the heterogeneities in material properties of strongly-bonded multi-material composite structures, there exist singularities of the stress fields at the material interface as well as the interface corners on the bounding surfaces. The finite-element method is probably the only plausible option to analyze the stress around such interfaces between dissimilar materials subjected to combined thermo-mechanical loading. In this article, we propose a novel mathematical model for the computational analysis of coupled thermo-elastic fields with perfectly bonded interfaces of dissimilar isotropic/orthotropic composite-material structures. A scalar function of the spatial variables, namely, the displacement function, is used to integrate the temperature field and the associated displacement-function field to obtain two nonhomogeneous partial-differential equations of equilibrium that govern two thermo-mechanical sub-problems defined in terms of the appropriate conditions of orthogonal temperature gradients. For a specific temperature field, obtained as a steady-state solution of the given thermal problem, the two thermo-mechanical sub-problems are solved independently to obtain the full-field solution of the thermo-elastic problem using the method of superposition. As an alternative to the conventional computational approach, we employed a single variable semi-analytical method of solution to determine the thermo-mechanical response of the multi-material composite structures. The continuity of the temperature and the heat flux in the thermal field problem and the continuity of the displacement and the traction vectors are utilized in the elastic field problem to determine the appropriate algebraic equations suitable for nodal application at the interfaces as well as interface corners on the boundaries of the respective problems. The quantitative comparison of the present displacement-function solutions to several mixed-boundary-value problems with those obtained by the conventional finite-element method verifies the high accuracy and effectiveness of the present method. This computational model has potential applications in identifying near-interface high-stress locations as well as possible critical regions of the physical boundary of composite structures subjected to mixed modes of thermal and mechanical loadings.

A liquid-solid-chemical coupled mass transport model for concrete considering phase assemblages and microstructure evolution

Concrete is a highly alkaline porous medium that is prone to chemical reactions with acidic environments. The ion transport coupled with chemical reactions between concrete and solutions in acidic environments (acid rain, acid groundwater, acid industrial water, etc.) is a major challenge to the durability design of such infrastructure. Here, a liquid–solid-chemical coupled mass transport model of concrete considering phase assemblages and microstructure evolution was developed to tackle this problem. The highly nonlinear evolution of phase assemblages was solved via chemical thermodynamics. Further analysis was conducted on the spatiotemporal relationship of microstructure evolution caused by reaction kinetics, and the dynamic porosity and ion adsorption capacity were taken into account in the mass transfer equations. Furthermore, the nonlinear non-homogeneous partial differential equations were solved via finite difference method. Finally, the proposed model was validated based on four different types of cement-based concrete. The results indicated that considering the chemical reaction between highly alkaline concrete and acidic environments was crucial for the transport of chloride ions. The migration depth varied up to twice under the influence of acidic solutions. This study builds a bridge between mass transfer and cement chemistry knowledge, and is expected to improve the accuracy of life prediction for reinforced concrete structures.

The meshless method of fundamental solutions applied to couple stress problems

In this paper, the meshless method of fundamental solutions is developed to solve couple stress problems even with the existence of body forces. Suitable new particular solutions are derived to deal with the non-homogeneous partial differential equations keeping the meshless essence of the developed technique. The derivation of the new particular solutions for generalizing displacements and tractions are discussed in detail. Several examples including stress concentration and problems with body forces are tested to demonstrate the efficiency and strength of the derived formulation.

Analysis of three-dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

In this work, we present a deep collocation method (DCM) for three-dimensional potential problems in non-homogeneous media. This approach utilizes a physics-informed neural network with material transfer learning reducing the solution of the non-homogeneous partial differential equations to an optimization problem. We tested different configurations of the physics-informed neural network including smooth activation functions, sampling methods for collocation points generation and combined optimizers. A material transfer learning technique is utilized for non-homogeneous media with different material gradations and parameters, which enhance the generality and robustness of the proposed method. In order to identify the most influential parameters of the network configuration, we carried out a global sensitivity analysis. Finally, we provide a convergence proof of our DCM. The approach is validated through several benchmark problems, also testing different material variations.

Stabilization of a class of underactuated Euler Lagrange system using an approximate model

The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.

On solving non-homogeneous partial differential equations with right-hand side defined on the grid

An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential equations (ODE system) with initial conditions at each grid point and includes the following sequence of actions. We seek a solution to the original equation, depending on one independent variable. The original equation is associated with a certain system of relations containing arbitrary functions and including the partial differential equation of the first order. For an equation of the first order, an extended system of equations of characteristics is written. Adding to it the remaining relations containing arbitrary functions, and demanding that these relations be the first integrals of the extended system of equations of characteristics, we arrive at the desired ODE system, completing the reduction. The proposed algorithm allows at each grid point to find a solution of the original partial differential equation that satisfies the given initial and boundary conditions. The algorithm is used to obtain solutions of the Poisson equation and the equation of unsteady axisymmetric filtering at the points of the grid on which the right-hand sides of the corresponding equations are given.

Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.

Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations

In this paper, the Laplace Differential Transform Method (LDTM) was utilized to solve some nonlinear nonhomogeneous partial differential equations. This technique is the combined form of the Laplace transform method with the Differential Transform Method (DTM). The combined method is efficient in handling nonlinear nonhomogeneous partial differential equations with variable coefficients. Laplace transform is introduced to overcome the inadequacy resulted from unsatisfied boundary condition in using DTM. Illustrative examples were examined to demonstrate the effectiveness of Laplace differential transform method. Results revealed that the LDTM is well appropriate for use in solving such problems.

Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomian Polynomial

In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomian polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.

On the existence of solutions to an inhomogeneous pantograph type equation with singular coefficients

We establish the existence of solutions to an initial boundary value problem that involves a certain class of nonhomogeneous functional partial differential equations of the pantograph type with singular time dependent coefficients. The problem is motivated from a cell division equation. The techniques for solving such problems are rare due to the nonlocal term and the singularity in coefficients.

Electromagnetohydrodynamic Electroosmotic Flow and Entropy Generation of Third-Grade Fluids in a Parallel Microchannel.

The present paper discusses the electromagnetohydrodynamic (EMHD) electroosmotic flow (EOF) and entropy generation of incompressible third-grade fluids in a parallel microchannel. Numerical solutions of the non-homogeneous partial differential equations of velocity and temperature are obtained by the Chebyshev spectral collocation method. The effects of non-Newtonian parameter Λ, Hartman number Ha and Brinkman number Br on the velocity, temperature, Nusselt number and entropy generation are analyzed in detail and shown graphically. The main results show that both temperature and Nusselt number decrease with the non-Newtonian physical parameter, while the local and total entropy generation rates exhibit an adverse trend, which means that non-Newtonian parameter can provoke the local entropy generation rate. In addition, we also find that the increase of non-Newtonian parameter can lead to the increase of the critical Hartman number Hac.

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

Unsteady Plasma Flow Near an Oscillating Rigid Plane Plate Under the Influence of an Unsteady Nonlinear External Magnetic Field

In an irreversible thermodynamics framework, Stokes’ second problem was examined for unsteady oscillating flow. The objective was to apply that for a plasma near an illimitable oscillating rigid plane plate under the influence of an unsteady nonlinear applied magnetic field. The Bhatnagar-Gross-Krook (BGK) pattern of the Boltzmann kinetic equation supplemented by Maxwell’s equations was investigated. The method of moments was applied with a two-sided distribution function. The exact traveling wave solution was obtained for a system consisting of four non-homogeneous partial differential equations. The velocity, shear stress, viscosity coefficient, generated electric field, applied nonlinear magnetic field, polarization, gyro-radius, and gyro-frequency were calculated. Furthermore, the distinction between the equilibrium velocity distribution function and the perturbed distribution functions was theoretically clarified at distinct time values. The advantage of the Boltzmann equation permitted us to consider irreversible non-equilibrium thermodynamics principles. For that purpose, the calculated distribution functions should be used in the formulae of entropy, entropy flux, thermodynamic forces, and kinetic coefficient. From the analysis of the results, it is found that Boltzmann’s H-theorem, thermodynamics laws, and Le Chatelier’s principle were consistent with our model for the whole system. The distinct contributions of the forces exerted on the system modified its internal energies; they were expressed via the Gibbs formula. The results demonstrated that the proposed model is capable of describing the behavior of plasma helium gas in the upper atmosphere ionic belts. Based on the analytical calculations, 3D-Graphics illustrating the physical quantities were drawn to predict their conduct, and the results are deeply discussed.

A New Approach for the Solution of Space-Time Fractional Order Heat-Like Partial Differential Equations by Residual Power Series Method

The main concern of this article has been to apply the Residual Power Series Method (RPSM) effectively to find the exact solutions of fractional-order space-time dependent nonhomogeneous partial differential equations in the Caputo sense. Our first step is to reduce fractional-order space-time dependent non-homogeneous partial differential equations to fractional-order space-time dependent homogeneous partial differential equations before applying the proposed method. Obtaining fractional power series solutions of the problem and reproducing the exact solution is the main step. The illustrative examples reveal that RPSM is a very significant and powerful method for obtaining the solution of any-order time-space fractional non-homogeneous partial differential equations in the form of fractional power series.

The differential-symbol method of constructing the quasipolynomial solutions of two-point in time problem for nonhomogeneous partial differential equation

The existence of the solution of nonhomogeneous partial differential equations (PDE) of second order in time and finite or infinite order in spatial variable with quasipolynomial right-hand side is proved. This solution satisfies the homogeneous two-point in time conditions. The differential-symbol method for constructing the solution of the problem is proposed. The examples of applying this method for solving some two-point problems for PDE are suggested.