Rational Solutions of Riccati-like Partial Differential Equations

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

A comparison of differential quadrature methods for the solution of partial differential equations

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

Transforming the nonlinear Black-Scholes equation into the diffusion PDE by introducing the log transform ofSand(T−t)→τcan provide the most stable platform within which option prices can be evaluated. The space jump that appeared in the transformation model is suitable to be solved by the sparse grid approach. An adaptive sparse approximation solution of the nonlinear second-order PDEs was constructed using Faber-Schauder wavelet function and the corresponding multiscale analysis theory. First, we construct the multiscale wavelet interpolation operator based on the definition of interpolation wavelet theory. The operator can be used to discretize the weak solution function of the nonlinear second-order PDEs. Second, using the couple technique of the variational iteration method (VIM) and the precision integration method, the sparse approximation solution of the nonlinear partial differential equations can be obtained. The method is tested on three classical nonlinear option pricing models such as Leland model, Barles-Soner model, and risk adjusted pricing methodology. The solutions are compared with the finite difference method. The present results indicate that the method is competitive.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, the partial differential equations which govern the steady-state heat transfer process in bodies with temperature-dependent thermal conductivity and temperature-dependent internal heat source. In addition, it will be considered a large class of nonlinear boundary conditions, especially those arising from the description of the heat exchange process from/to bodies at high temperature levels. Proofs of the solution’s existence and uniqueness are presented.

On the numerical solution of partial differential equations with two spatial scales

In this paper, we use the hybridization of two well-known semi-analytical methods to obtain the numerical solutions for some special linear and nonlinear partial differential equations that are predominant in most physical science disciplines. The first among these methods is the coupling of Laplace transform and Adomian decomposition method (LADM) while the second method is standard Homotopy perturbation method with Laplace transform method (HPTM). The accuracy and dependability of these proposed techniques is confirmed by applying them to solve linear and nonlinear Kleidon-Gordon equations, linear transverse equation of a vibrating beam, homogeneous and inhomogeneous nonlinear PDEs, advection equation, diffusion-convection and Korteweg-DeVries equation. Thereafter, comparison between the solutions obtained by the methods is presented in tables for convergence analysis. Consequently, the findings from our study showed the two methods can be effective alternative approaches for obtaining solutions to linear and nonlinear PDEs and higher-order initial value problems.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Numerical convective schemes based on accurate computation of space derivatives

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our earlier studies of analytic solutions. The proof relies on the result obtained by Denef and Lipshitz concerning the relationship between a certain subset of nonhomogeneous differential equations of the considered form (but without boundary conditions) and Diophantine equations.

L iSS — An environment for the parallel multigrid solution of partial differential equations on general 2D domains

Solution of the partial differential equation of the Hodgkin-Huxley model using differential quadrature