An efficient numerical algorithm for the study of time fractional Tricomi and Keldysh type equations

Engineering with Computers - This work addresses a hybrid scheme for the numerical solutions of time fractional Tricomi and Keldysh type equations. In proposed methodology, Haar wavelets are used...     

An efficient numerical algorithm for multi-dimensional time dependent partial differential equations

An efficient and robust numerical scheme based on Haar wavelets and finite differences is suggested for the solution of two-dimensional time dependent linear and nonlinear partial differential equations (PDEs). Excellent feature of the scheme is the conversion of linear and non-linear PDEs to algebraic equations which are comparatively easy to handle. Convergence of the scheme, which guarantees small error norm as the resolution level increases, is also an important part of this work. Different error norms are computed to check efficiency of the technique. Computations verify accuracy, flexibility and low computational cost of the method.     

A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $L_2,L_{\infty }$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).     

Numerical solutions of two dimensional Sobolev and generalized Benjamin–Bona–Mahony–Burgers equations via Haar wavelets

This work addresses, numerical method based on Haar wavelets and finite differences to solve two dimensional linear, nonlinear Sobolev and non-linear generalized Benjamin–Bona–Mahony–Burgers (NGBBMB) equations. The temporal part is discretized using finite differences while spatial part is approximated by two dimensional Haar wavelets. With this strategy, computing solution of two dimensional PDEs reduces to computing solution of linear system of algebraic equations. Collocation approach is then applied to determine the wavelet coefficients from linear system. This paper shows that two dimensional Haar wavelets are suitable and effective for two dimensional linear and non-linear PDEs. For validation of the proposed scheme different problems have been solved and error norms $L_{\infty },L_2$ are computed. Computation verifies that current scheme has good outcome.     

Solution of strongly nonlinear ordinary differential equations arising in heat transfer with optimal homotopy asymptotic method

An analytic approximate procedure, called optimal homotopy asymptotic method (OHAM) is considered for the solution of strongly nonlinear differential equations arising in heat transfer. Two particular applications are chosen: convecting-radiating cooling of a lumped system with variable specific heat and fins with temperature dependent surface heat flux. The effectiveness of the method, which is independent of the small parameter, is investigated by comparing the results obtained with the exact and the results already present in the literature.     

A grid-based assessment of global water scarcity including virtual water trading

A 0.5-degree grid-based assessment of the scarcity of global water resources including virtual water trading has been made. The three components of water availability considered for each grid were local runoff, routed flow from upstream and virtual water trading. Several assumptions were postulated to convert country-base estimations of virtual water trading to grid values. The results show that unequal spatial distribution of global water resources had been considerably neutralized by virtual water trading. A large proportion of people in the Middle-East, North-Africa and Sub-Sahara region are able to relieve their water stress through virtual water import. The paper also reports two hypothetical scenarios with extremes of natural flow availability based on the presence and absence of routed upstream flow.     

The solution of multipoint boundary value problems by the Optimal Homotopy Asymptotic Method

In this work, we solve multipoint boundary value problems using the Optimal Homotopy Asymptotic Method (OHAM).The proposed method is tested upon several problems from the literature and the results are compared with the available exact solution. This method provides easy tools to control the convergence region of approximating solution series where ever necessary.     

Development and Application of an Integrated Water Balance Model to Study the Sensitivity of the Tokyo Metropolitan Area Water Availability Scenario to Climatic Changes

This study investigates the water availability scenario along Tokyo Metropolitan Area (TMA) under future climatic changes. TMA, which depends largely on the adjacent Tone river for its water supply, suffers from water shortage almost once in 2–3 yr. A methodology was developed considering integrated approach to ascertain probable impact of climatic changes on the overall water availability along the Tone river and its impact on TMA. Historical trend in hydro-climatic characteristics of the Tone River basin was investigated at the first place to assess the changes and interrelationships. A deterministic water balance model was later developed, integrating natural hydrological balance as well as several water uses and river regulation effects. The natural water balance part of the model was tested for some hypothetical climate change scenarios to observe the sensitivity of the Tone river flow to climatic perturbations. For the worst scenario of precipitation and temperature changes, the unregulated Tone river flow was observed to be reduced by around 20–50%, varying over different months. After considering river regulation effects from reservoirs and diversion, the regulated flow was further analyzed based on some risk indices. Drought risk was observed to be increased significantly for the periods between April–July.     

Fabrication of and performance studies on a low cost solar cooker having an inclined aperture plane

A low cost box-type solar cooker made of two paper carton boxes with crumpled newspaper balls as insulation has been fabricated with a tilted aperture plane. Comparative tests of this cooker have been conducted against a normal type costlier solar cooker with 1000 ml load of water in each of the cookers. It has been observed that on a sunny day water temperature initially increases more rapidly in the new cooker compared to the normal type cooker. But at temperatures higher than 90°C both the cookers perform similarly. Two figures of merit F₁ and F₂ have also been found to be satisfactory. An arrangement of low cost auxiliary heating using a 100w electric bulb inside a blackened metal casing allows the carton box cooker to reach cooking temperatures under cloudy conditions with ease when the normal type solar cooker fails. The cost of materials for the new cooker is within US $10 and has been observed to be as effective as the normal type solar cooker which requires between US $40 and $65 to fabricate.