The Fundamental Differential Form and Boundary-Value Problems

A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE’s) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE’s and several of its variations such as the Wiener–Hopf technique. (b) The integral representation of the solution of linear PDE’s in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.

Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM)

On the smoothing property of linear delay partial differential equations

A new integrable nonlinear partial differential equation (PDE) in 2+1 dimensions is derived starting from the Konopelchenko–Dubrovsky equation. We use an asymptotically exact reduction method based on Fourier expansion and spatio-temporal rescaling and obtain a new integrable Davey–Stewartson-type equation. In order to demonstrate the integrability of the new equation by the inverse scattering method, we apply the reduction technique to the Lax pair of the Konopelchenko–Dubrovsky equation and find the corresponding Lax pair of the new equation. The new equation reduces to the Davey–Stewartson or the nonlinear Schrodinger equation by appropriate limits.

A novel heuristic technique that enhances the search facility of the standard genetic programming (GP) algorithm is presented. The method provides a dynamic sniffing facility to optimize the local search in the vicinity of the current best chromosomes that emerge during GP iterations. Such a hybrid approach, that combines the GP method with the sniffer technique, is found to be very effective in the solution of inverse problems where one is trying to construct model dynamical equations from either finite time series data or knowledge of an analytic solution function. As illustrative examples, some special function ordinary differential equations (ODEs) and integrable nonlinear partial differential equations (PDEs) are shown to be efficiently and exactly recovered from known solution data. The method can also be used effectively for solution of model equations (the direct problem) and as a tool for generating multiple dynamical systems that share the same solution space.

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.

The solution of the Cauchy problem, an infinite set of conserved quantities, a variational formulation, the auto-Backlund transformations and the soliton phenomenology (featuring singly and doubly humped solitons, as well as breathers) are surveyed briefly for the integrable non-linear evolution partial differential equation (PDE) ut+(uxx-2 eta u3-23uux3/( eta +u2))x=0, eta =+or-1.

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

In shallow water theory, completely integrable nonlinear partial differential equations arise at various levels of approximations/simulations. Such equations possess traveling wave solutions, called solitons, which themselves are self-phase-modulated signals. Wavelets are traditionally applied in signal processing but it has limitations to deal effectively with geometric features with line and surface singularities. Due to its poor orientation selectivity, wavelet can only capture limited directional information in signal processing. A multi-resolution geometric analysis, named curvelet transform overcomes the drawbacks of the conventional wavelet transform. Such a geometric characterization provides better understanding of cascade mechanism and dissipation range dynamics. In the present work, a multi-scale geometric analysis with curvelets as basis functions is proposed for analyzing higher dimensional lump-like solitons- that arise in the evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromigration equation. Moreover, the application of curvelet thresholding technique for de-noising the contaminated signals is presented.

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev–Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann–Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.

We study two models which describe subdiffusive processes. Subdiffusion is defined by the relation \(\left \langle (\Delta x)^{2}(t)\right \rangle = D_{\alpha }t^{\alpha }\), where \(\left \langle (\Delta x)^{2}(t)\right \rangle\) denotes the mean square displacement, α is a subdiffusion parameter which obeys 0 < α < 1 and D α is a subdiffusion coefficient. The first model consists of a nonlinear partial differential equation with derivatives of a natural order obtained from a Sharma–Mittal nonadditive entropy, whereas the second model is based on a linear partial differential equation with a fractional time derivative which is derived from the continuous time random walk formalism. We obtain the fundamental solutions for both models. Next, we assume two agreement conditions. According to the first one the fundamental solutions for both model fulfill the relation which defines subdiffusion. The second agreement condition presumes the equality of the first passage time distributions. On the basis of these agreement conditions we answer the question when the considered models are similar.KeywordsSubdiffusionFractional derivativeNonadditive entropy

The Fourier spectral method which uses the Fourier transform and the Fourier inverse transform are applied to the nonlinear and linear partial differential equations. The partial differential equation becomes the ordinary differential equation after the Fouier transform used and it is solved by the Runge-Kutta method. The solutions of linear Burgers' equation are carefully checked by being compared with the analytical solutions and this method was applied to nonlinear Burgers' equation. The initial conditions are cos x, sin x and Gaussian distribution.

The aim of this research is to study the Z transformation to solve linear partial difference equations. This is due to the fact that linear partial difference equations cannot be solved directly using the Z transformation, and the research concluded that direct partial difference conditions cannot be treated directly by the Z transformation except after converting the partial difference equations to linear difference equations using one of the transformation methods. The research recommended using Binomial transformation to solve partial difference equations.