Abstract
AbstractIn this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameterεand gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.
Highlights
Perturbed partial differential equations arise in a wide variety of application fields such as biosciences, economics, material science, medicine, robotics etc. [5, 14, 20] and in the last few decades there has been a growing interest in the study of delay differential equations [3, 4, 9]
The excitatory input contributes to the membrane potential by amplitude as with intensity λs and the inhibitory input contributes by amplitude is with intensity ωs
Bansal et al.[3, 4] developed parameter uniform numerical schemes to find the approximate solution of time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments in space variable, which work for small shifts as well as large shifts
Summary
Perturbed partial differential equations arise in a wide variety of application fields such as biosciences, economics, material science, medicine, robotics etc. [5, 14, 20] and in the last few decades there has been a growing interest in the study of delay differential equations [3, 4, 9]. Bansal et al.[3, 4] developed parameter uniform numerical schemes to find the approximate solution of time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments in space variable, which work for small shifts as well as large shifts. If we solve singularly perturbed partial differential equations using central finite difference scheme on a uniform mesh, it gives oscillatory solution, which shows that method is unstable To deal with such situation, more mesh points in boundary layer region is required. The major drawback of Shishkin meshes is the requirement of prior information of the location of the layer regions To overcome this drawback, in this paper, we proposed an adaptive mesh method using the concept of entropy function for solving convection-diffusionreaction singularly perturbed delay parabolic partial differential equations. The problem (1) with initial condition and the interval boundary conditions (2), can be rewritten as
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