The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

SOLVING COUPLED PSEUDO-PARABOLIC EQUATION USING A MODIFIED DOUBLE LAPLACE DECOMPOSITION METHOD

In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.

Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.

Comparison between the Laplace Decomposition Method and Adomian Decomposition in Time-Space Fractional Nonlinear Fractional Differential Equations

The aim of this paper is to discuss application of Laplace Decomposition Method with Adomian Decomposition in time-space Fractional Nonlinear Fractional Differential Equations. The approximate solutions result from Laplace Decomposition Method and Adomian decomposition;those two accessions are comfortable to perform and firm when to PDEs. For caption and further representation of the thought, several examples are tool up.

Solving <i>nth</i>-Order Integro-Differential Equations Using the Combined Laplace Transform-Adomian Decomposition Method

In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF INHOMOGENEOUS p-LAPLACE EQUATION

Let Ω be a smooth bounded domain in R^n. In this article, we consider the homogeneous boundary Dirichlet problem of inhomogeneous p-Laplace equation --△pu = ｜u｜^q-1 u ＋ λf（x） on Ω, and identify necessary and sufficient conditions on Ω and f（x） which ensure the existence, or multiplicities of nonnegative solutions for the problem under consideration.

Formal Similarity of Solutions in the Laplace Space on the Class of Fluid Flow Differential Equation

In this paper, the fluid flow differential equation based on the homogenous reservoirs model is first reviewed. Then a theorem about the formal similarity of solutions in the Laplace space with outer boundary conditions and inner boundary condition is presented and proved. Lastly, a corollary of our theorem is given particularly on inner boundary. The obtained results are very helpful for understanding inherent laws of relevant engineering science and designing practical analysis software.

p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL

In this paper, we deal with the following problem：By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when

STRUCTURE OF NONNEGATIVE NONTRIVIAL AND POSITIVE SOLUTIONS OF SINGULARLY PERTURBED p-LAPLACE EQUATIONS

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.

Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform

We discuss the solution of Laplace’s differential equation and a fractional differential equation of that type, by using analytic continuations of Riemann-Liouville fractional derivative and of Laplace transform. We show that the solutions, which are obtained by using operational calculus in the framework of distribution theory in our preceding papers, are obtained also by the present method.

HIGH-ORDER BOUNDARY CONDITIONS FOR THE PROBLEMS OF LAPLACE EQUATION IN INFINITE REGION AND THEIR APPLICATION

The high-order boundary conditions for the problems cf Laplace equation in infinite region have been developed. The improvement in accuracy for numerical solution is achieved by imposing the high-order boundary conditions on the exterior boundarv of a reduced finite region in which the numerical method is used. So both the computing efforts and the required storage in computer are reduced. The numerical examples show that the 1st-order boundary condition approaches to the exact boundary condition and it is clearly superior to the traditional boundary condition and the 2nd-order boundary condition.

Exact solutions of the Klein-Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angulm＂ functions are expressed in terms of the hypergeometric functions. The radial eigenfunetions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.

Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.

UNIFORM MEYER SOLUTION TO THE THREE DIMENSIONAL CAUCHY PROBLEM FOR LAPLACE EQUATION

We consider the three dimensional Cauchy problem for the Laplace equation{uxx（x,y,z）＋uyy（x,y,z）＋uzz（x,y,z）=0,x∈R,y∈R,0〈z≤,u（x,y,0）=g（x,y）x∈R,y∈R,uz（x,y,0）=0,x∈R,y∈R,where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 〈 z 〈 1. The problem is ill-posed, the solution （if it exists） doesn＇t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.

A Computational Quadruple Laplace Transform for the Solution of Partial Differential Equations

In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homogeneous as well as non-homogeneous partial differential equations involving four variables. The performance of quadruple Laplace transform is shown to be very encouraging by concrete examples. An elementary table of quadruple Laplace transform is also provided.

Solution of the Third Kind Boundary Value Problem of Laplace’s Equation Based on Conformal Mapping

In order to overcome the difficulty in solving the boundary value problem of electrostatic field with complex boundary and to give a new method for solving the third boundary value problem of Laplace’s equation, in this paper, the third boundary value problem of Laplace’s equation is studied by combining conformal mapping with theoretical analysis, the several analytical solutions of third boundary value problems of Laplace’s equation are gives, the correctness of its solution is verified through computer numerical simulation, and a new idea and method for solving the third boundary value problem of Laplace’s equation is obtained. In this paper, the boundary condition of the solving domain is changed by the appropriate conformal mapping, so that the boundary value problem on the transformed domain is easy to be solved or be known, and then the third kind boundary value of the Laplace’s equation can be solved easily;its electric potential distribution is known. Furthermore, the electric field line and equipotential line are plotted by using the MATLAB software.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Laplace Transform Method Applied to Solve Fractional Difference Equations

In this paper, we discuss the Laplace transform of the Caputo fractional dierence and the fractional discrete Mittag-Leer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.

Laplace Discrete Adomian Decomposition Method for Solving Nonlinear Integro Differential Equations

This paper proposes the Laplace Discrete Adomian Decomposition Method and its application for solving nonlinear integro-differential equations. This method is based upon the Laplace Adomian decomposition method coupled with some quadrature rules of numerical integration. Four numerical examples of integro-differential equations in both Volterra and Fredholm integrals are used to be solved by the proposed method. The performance of the proposed method is verified through absolute error measures between the approximated solutions and exact solutions. The series of experimental numerical results show that our proposed method performs in high accuracy and efficiency. The study clearly highlights that the proposed method could be used to overcome the analytical approaches in solving nonlinear integro-differential equations.

An Analysis of Double Laplace Equations on a Concave Domain

In the investigation, the complex geometric domain is a concave geometrical pattern. Due to the symmetric character, the left side of the geometric pattern,?i.e.?the L-shaped region is calculated in the study. The governing equation is expressed with?Laplace?equations. And the analysis is solved by eigenfunction expansion and point-match method. Besides, visual?C++?helps obtain the results of numerical calculation. The local values and the mean values of the function are also discussed in this study.