Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

Discrete (G'/G )-expansion: a Method Used to Get Exact Solution of Fdde (Fractional Differential-difference Equation) Linked With Nltl (Non-linear Transmission Line)

Here, we have used the discrete (G'/G)-expansion procedure with the derivative operator MR-L (modified Riemann-Liouville) and FCT (fractional complex transform) to find the exact/analytical solution of an electrical transmission line which is non-linear. Results include solutions for integer and fractional DDE. We consider two special cases of solutions: hyperbolic and trigonometric. Hyperbolic solutions indicate propagation of singular wave on the transmission line. Trigonometric solutions show propagation of complex wave.

Quartic Non-Polynomial Spline Method for Singularly Perturbed Differential-difference Equation with Two Parameters

Quartic non-polynomial spline method is presented to solve the singularly perturbed differential-difference equation containing two parameters. The considered equation is transformed into an asymptotical equivalent differential equation, and the derivatives are replaced finite difference approximation using the quartic non-polynomial spline method. The convergence analysis of the method has been established. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Furthermore, the present method gives a more accurate solution than some existing methods reported in the literature.

Analytical Construction of Linear Loop Response with Delay Covered by Feedback

Disclosed is a method of obtaining an analytical description of a response of a linear chain with a lag covered by feedback. The process is considered at consecutive intervals. Their duration is equal to the time of net lag. The possibility of obtaining an analytical description of the process is achieved by solving the linear differential equation of the open circuit, and not the differential-difference equation of the closed circuit. An example of analytical construction of the response of the simplest chain with a delay covered by negative feedback is given, and its digital modeling is carried out. The coincidence of the results of the calculation and digital modeling of the circuit confirmed the correctness of the proposed method of analytical response construction.

Results on the solutions of several second order mixed type partial differential difference equations

<abstract><p>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} </tex-math></disp-formula></p> <p>where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi ^{[<xref ref-type="bibr" rid="b23">23</xref>]}, Xu and Cao ^{[<xref ref-type="bibr" rid="b35">35</xref>]}, Liu, Cao and Cao ^{[<xref ref-type="bibr" rid="b17">17</xref>]}.</p></abstract>