A solution for LAPLACE partial differential EQUATION by using Spline basis functions is presented. The formulation is derived and its differences with the finite element method are explained. The effect of some of parameters such as the knot vector and grid of control points on the solution is investigated. Finally, a few examples are presented to demonstrate the efficiency of the method.

In this work we constructed a numerical scheme to approximate the Volterra integro-differential EQUATIONs of convolution type using LAPLACE transform. The solution of the problem is recovered using inverse LAPLACE transform as contour integral in the complex plane. The integral is then approximated along a suitable contour using the trapezoidal rule with equal step size. The solution accuracy depends on optimal contour of integrations to compute accurately the inverse LAPLACE transform. For better accuracy two types of contour parabolic and hyperbolic are used which are available in the literature. The performance of the numerical scheme is tested for different examples. The actual error well agree with the corresponding error estimates of the proposed numerical scheme for both parabolic as well as hyperbolic contours.

In this study, the homotopy perturbation transform method (HPTM) is performed to give analytical solutions of the time fractional diffusion EQUATION. The HPTM is a combined form of the LAPLACE transform and homotopy perturbation methods. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is very effective and simple in performing a solution to the fractional partial differential EQUATION. A solution has been plotted for different values of a, and some numerical illustrations are given.

We study the Schr\"odinger EQUATION   $\left(\mathrm{Q}_{\varepsilon}\right)$: $- \varepsilon^{2(p-1)} \Delta_p v + V(x)\, |v|^{p-2} v - |v|^{q-1}v = 0$, $x \in \mathbb{R}^N$, with $v(x) \rightarrow 0$ as $|x| \rightarrow+\infty$, for the infinite case, as given by Byeon and Wang for a situation of critical frequency,  $\displaystyle \{x\in \mathbb{R}^N \, / \: V(x) = \inf V = 0\} \neq \emptyset$. In the semiclassical limit, $\varepsilon \rightarrow 0$, the corresponding limit problem is $\left(\mathrm{P}\right)$: $\Delta_p w+|w|^{q-1} w=0$, $x \in \Omega$, with $w(x)=0, x \in \partial \Omega$, where $\Omega \subseteq \mathbb{R}^N$ is a smooth bounded strictly star-shaped region related to the potential $V$. We prove  that for $\left(\mathrm{Q}_{\varepsilon}\right)$ there exists a non-trivial solution with any prescribed $\mathrm{L}^{q+1}$-mass.Applying a Ljusternik-Schnirelman scheme, shows  that  $\left(\mathrm{Q}_{\varepsilon}\right)$ and $\left(\mathrm{P}\right)$ have infinitely many pairs of solutions. Fixed a topological level $k \in \mathbb{N}$, we show that a solution of $\left(\mathrm{Q}_{\varepsilon}\right)$, $v_{k, \varepsilon}$, sub converges, in $\mathrm{W}^{1,p}(\mathbb{R}^N)$ and up to scaling, to a corresponding solution of $\left(\mathrm{P}\right)$. We also prove that the energy of each solution, $v_{k,\eps}$ converges to the corresponding energy of the limit problem  $\left(\mathrm{P}\right)$ so that the critical values of the functionals associated, respectively, to  $\left(\mathrm{Q}_{\varepsilon}\right)$ and $\left(\mathrm{P}\right)$ are topologically equivalent.

In this paper we formulate a LAPLACE-transform multiple scale expansion procedure to develop asymptotic solution of weakly non-linear partial differential EQUATION. The method is applied to some general nonlinear wave and diffusion EQUATIONs.

Rivers are one of the most important natural water resources in the world. Pollution transport modeling in rivers is performed by the partial advection-dispersion-reaction EQUATION (ADRE). In the present study, using the LAPLACE transform, which is a powerful and useful tool in solving differential EQUATIONs, the analytical solution of the ADRE EQUATION was obtained in a finite domain with variable coefficients for the upstream and downstream Dirichlet boundary conditions and the initial zero condition in the river. To use the analytical solution in this study, three examples are presented, each of which, the river are divided into two, four, and eight parts, which, while the parameters of flow, pollution, and river geometry are variable in all three examples, for each of the examples, the accuracy of the analytical solution available when the segmentation of the intervals increases as compared to the numerical solution. By specifying the matrices of velocity, dispersion coefficient, cross-section, etc. as input to the problem, the diffusion matrix is calculated and, consequently, a complex system of EQUATIONs is created that doubles the complexity of the work. The amount of pollutant concentration is calculated by solving the system of the above EQUATIONs. The numerical solution is used to validate the existing analytical solution, the results showed that the greater the number of river divisions, the higher the accuracy of the solution, and the two analytical and numerical solutions will be well compatible with each other. Given the ability and performance of the existing analytical solution, it can be acknowledged that the analytical solution in this study can be used as a tool to validate and verification numerical solutions and other analytical solutions for the coefficients of the EQUATION.

Alternative specifications of univariate asymmetric LAPLACE models are described and investigated. A more general mixture model is then introduced. Bivariate extensions of these models are discussed in some detail, with particular emphasis on associated parameter estimation strategies. Multivariate versions of the models are briefly introduced.