In this paper, we consider the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system of equations. This system of equations is a generalization of the cmKdV equation in the (2+1)-dimension and has great significance in the fields of applied magnetism and nanophysics. On the basis of the Lax pair, infinitely many conservation laws are obtained. In addition, the multi-waves, homo clinic breather, rational, and interactions solutions of this equation are derived with the aid of logarithmic transformation and symbolic computation. For the suitable value of parameters, the 3D surfaces of obtained solutions have been plotted using Mathematica.
In this paper, a nonlinear eigenvalue problem consisting of a nonlinear Sturm-Liouville equation -y ''- q(x)y =lambda q(-1)(x)y (R) with Dirichlet boundary conditions on the interval (-1/2, 1/2) is investigated, where lambda > 0 is the eigenparameter. We provide a simple scheme to obtain the asymptotic behavior of L-& ell;-bifurcation curve lambda = lambda(& ell;)(gamma) as gamma -> 0, where gamma =||y(lambda)||& ell;, & ell;>= 1, and y(lambda) is the solution of Dirichlet problem associated with lambda.
This paper examines the solutions of nonlinear higher-order singular Emden-Fowler type equations arising in various physical models. Generally, it becomes difficult to obtain the solution near the point of singularity. To overcome this problem, an iterative technique is introduced that depends on the variational iteration method (VIM) and the homotopy perturbation method (HPM). Such a technique generates the solution in terms of a series, which is highly practical from computing perspective. An equivalent recursive integral representation (involving Lagrange's multiplier) for the higher-order nonlinear singular Emden-Fowler type (SEFT) equations with initial conditions (ICs) is established with the support of the variational iteration method (VIM). Making use of the concept of homotopy, a system of integral equations is established, which helps to deal with nonlinearity. Some numerical examples are studied through the proposed iterative technique to show the applicability and efficiency of the technique.
In this article, we present a highly accurate technique for the numerical solution of the variable-order time-fractional Burgers-Huxley equation. The original equation is first discretized in the temporal and spatial directions. The third-order weighted-shifted Gru & uml;nwald-Letnikov and the fourth-order compact finite difference methods are used. We then formulate a nonlinear system of algebraic equations using the fully discretized version of the problem. The derived nonlinear system is solved by utilizing an iterative algorithm. The analysis of solvability, stability, and convergence of the method is also addressed. The method achieves a convergence rate of four in the spatial direction and three in the temporal direction. Moreover, it is a low-cost computational method and easy to implement. Finally, various illustrative examples are solved to verify the accuracy of the proposed method.
The need for accurate solutions to mathematical models, particularly for linear and nonlinear higher-order initial value problems, is essential across various scientific and engineering fields. Traditional methods often face challenges with stability and precision, especially in non-linear cases, prompting the development of advanced numerical techniques. This study introduces a two-step overlapping adaptive step-size multi-derivative hybrid block method to address these challenges in solving higher-order initial value problems. The method incorporates overlapping elements, using the second-to-last intra-step point from the previous step within each integration block to enhance accuracy. The method uses error estimation and selects an appropriate step-size, ensuring the desired accuracy without wasting computational resources or introducing unnecessary errors. The non-linear initial value problems are efficiently linearized using a modified-Picard iteration. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method, and its performance is compared against a similar non-overlapping method as well as other methods reported in the literature.
