A Direct Matrix Method for Computing Analytical Jacobians of Discretized Nonlinear Integro-differential Equations

In this paper, some adaptive single-step methods like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), and Radau IIA (Rad) methods are implemented in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs). Maple's robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and the analytic Jacobian. The algorithm and implementation were found to be robust and efficient for index-1 DAE problems and scal...

A new method of numerical solution for partial differential equations is proposed. The method is based on a fast matrix multiplication algorithm. Two-dimensional Poison equation is used for comparison of the proposed method with conventional numerical methods. It was shown that the new method allows for linear growth in the number of elementary addition and multiplication operations with the growth of grid size, as contrasted with quadratic growth necessitated by the standard...

A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established identity which links symmetries of the underlying discrete equations, solutions of the discrete adjoint equations and first integrals. The method is applied to invariant mappings and discretizations of a second order and a third order ODEs. In ex...

By the theory of pseudoinverse matrices and orthoprojectors, we establish a criterion for the solvability and find the general form of solutions of an integrodifferential equation with with impulse action and control. The general form of control for which these solutions exist is also determined. 9 pages. The publication contains the research results of project No.2020.02/0089 with the grant support of the National Research Fund of Ukraine.

Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi-step integration methods. The coefficients are reverse engineered based on samples from a target function and its derivative used for training. The Runge-Kutta schemes are trained using the order condition equations. An appealing feature of the evolutionary m...

Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear system. In this way, all nonlinear algebraic terms are instead expressed as Linearly independent variables. Therefore, a n-dimension nonlinear system can be expanded as a linear system of n(n+1)/2 dimension space. This introduces the possi...

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis...

Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as finite differences. For complex applications, hand-coding the Jacobian can be time-consuming and error-prone, yet computationally efficient. Whilst finite difference approximations are straight-forward to implement, they have high arithmeti...

The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for $p$-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated ...

This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numericaloperators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chai...