Differential equations occur in many fields of science, engineering and social science as it is a natural way of modeling uncertain dynamical systems. A bipolar fuzzy set model is useful mathematical tool for addressing uncertainty which is an extension of fuzzy set model. In this paper, we study differential equations in bipolar fuzzy environment. We introduce the concept gH-derivative of bipolar fuzzy valued function. We present some properties of gH-differentiability of bipolar fuzzy valued function by considering different types of differentiability. We consider bipolar fuzzy Taylor expansion. By using Taylor expansion, Euler method is presented for solving bipolar fuzzy initial value problems. We discuss convergence analysis of proposed method. We describe some numerical examples to see the convergence and stability of the method and compute global truncation error. From numerical results, we see that for small step size Euler method converges to exact solution.

In this paper, we originate results with finite difference schemes to approximate the solution of the classical Fisher Kolmogorov Petrovsky Piscounov (KPP) equation from population dynamics. Fisher's equation describes a balance between linear diffusion and nonlinear reaction. Numerical example illustrates the efficiency of the proposed schemes, also the Neumann stability analysis reveals that our schemes are indeed stable under certain choices of the model and numerical parameters. Numerical comparisons with analytical solution are also discussed. Numerical results show that Crank Nicolson and Richardson extrapolation are very efficient and reliably numerical schemes for solving one dimension fisher's KPP equation.

Abstract. In this paper, we suggest and analyze two new algorithm of fourth and fifth order convergence. We rewrite nonlinear equation as an equivalent coupled system and then use modified decomposition technique to develop our algorithms. Convergence analysis of newly introduced algorithms has been discussed. To see efficiency and performance of these algorithms, we have made comparison of these algorithms with some well known algorithms existing in literature.

Dendrites can severely impair zinc battery performance. An in-depth understanding of the dynamic morphology evolution of dendrites with operando approaches is pivotal when addressing these issues. However, in previous studies, the corresponding electrochemical signals are usually ensemble and averaged. It is very challenging to obtain detailed information about the key morphology-performance relationship. Herein, correlated high-resolution operando optical and electrochemical studies of single dendrites on Pt reflective nanopore electrodes are reported. The zinc deposi-tion and dissolution can be directly imaged by a high NA optical microscope, while corresponding galvanic charging and discharging curves are obtained. The correlated information of morphology changes and dynamic overpotential fluctuations under different circumstances unveils the competition between active growth vs. passivation. The isolated zinc formation at the single dendrite level is also evaluated. The methodology can be further extended to elucidate the direct relationship between dendrite evolution and electrochemical responses in various battery systems.

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of 2 L , L ∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.