MODEL MATEMATIKA WAKTU PENGOSONGAN TANGKI AIR

Sebuah tangki diisi air dengan ketinggian tertentu dan pada dasar tangki tersebut dibuat lubang kecil, sehingga air mengalir melalui lubang tersebut. Menurut Hukum Torricelli, apabila tangki dikosongkan maka kecepatan keluar air akan berubah secara kontinu dipengaruhi oleh ketinggian air. Lamanya waktu yang digunakan untuk mengosongkan tangki apabila diisi air dengan ketinggian tertentu disebut waktu pengosongan. Waktu pengosongan juga dipengaruhi oleh bentuk tangki. Tulisan ini membahas tentang depletion ratio. Seandainya kecepatan keluar air adalah tetap, maka depletion ratio adalah perbandingan antara waktu pengosongan jika kecepatan keluar air berubah dipengaruhi oleh ketinggian air dengan waktu pengosongan jika kecepatan air keluar tetap. Selanjutnya jika diketahui depletion rationya maka dapat ditentukan bentuk geometri tangki tersebut

SUATU CONTOH INVERSE PROBLEMS YANG BERKAITAN DENGAN HUKUM TORRICELLISUATU CONTOH INVERSE PROBLEMS YANG BERKAITAN DENGAN HUKUM TORRICELLI

Makalah ini membahas tentang inverse problems yang berkaitan dengan hukum Torricelli. Jika suatu tangki berisi air yang pada bagian dasarnya terdapat sebuah lubang kecil, maka kecepatan air keluar dari lubang berubah sesuai dengan perubahan ketinggian air dalam tangki. Diberikan dua contoh inverse problems yang melibatkan hukum Torricelli. Berbeda dengan direct problems yang selalu menghasilkan solusi yang tunggal dan stabil, suatu inverse problems dapat mempunyai solusi yang tidak tunggal dan tidak stabil. Makalah ini tidak hanya membahas bagaimana menyelesaikan suatu inverse problems tetapi juga bagaimana sifat solusinya

The Combinational Mutation Strategy of Differential Evolution Algorithm for Pricing Vanilla Options and Its Implementation on Data during Covid-19 Pandemic

Investors always want to know about the profit and the risk that they will be get before buying some assets. Our main focus is getting the profit and the probability of getting that profit using the differential evolution algorithm for vanilla option pricing on data before and during COVID-19 pandemic. Therefore, we model the pricing of an option using a bi-objective optimization problem using data before and during COVID-19 pandemic for one year expiration date. We change this problem into an optimization problem using adaptive weighted sum method. We use metaheuristics algorithm like Differential Evolution (DE) algorithm to solve this bi-objective optimization problems. In this paper, we also use modification of Differential Evolution for getting Pareto optimal solutions on vanilla option pricing for all contract. The algorithm is called Combinational Mutation Strategy of Differential Evolution (CmDE) algorithm. The results of our algorithm are satisfactory close to the real option price in the market data. Besides that, we also compare our result with the Black-Scholes results for validation. The results show that our results can approximate the real market options more accurate than Black-Scholes results. Hence, our bi-objective optimization using Combinational Mutation Strategy of Differential Evolution algorithm can be used to approximate the market real vanilla option pricing before and during COVID-19 pandemic

Approximate Solutions of Linearized Delay Differential Equations Arising from a Microbial Fermentation Process Using the Matrix Lambert Function

In this paper we present approximate solutions of linearized delay differential equations using the matrix Lambert function. The equations arise from a microbial fermentation process in a metabolic system. The delay term appears due to the existence of a rate-limiting step in the fermentation pathway. We find that approximate solutions can be written as a linear combination of the Lambert function solutions in all branches. Simulations are presented for three cases of the ratio of the rate of glucose supply to the maximum reaction rate of the enzyme that experienced delay. The simulations are worked out by taking the principal branch of the matrix Lambert function as the most dominant mode. Our present numerical results show that the zeroth mode approach is quite reliable compared to the results given by classical numerical simulations using the Runge-Kutta method

Finding Multiple Solutions of Multimodal Optimization Using Spiral Optimization Algorithm with Clustering

Multimodal optimization is one of the interesting problems in optimization which arises frequently in a widerange of engineering and practical applications. The goal of this problem is to find all of optimum solutions in a single run. Some algorithms fail to find all solutions that have been proven their existence analytically. In our paper [1], a method is proposed to find the roots of a system of non-linear equations using a clustering technique that combine with Spiral Optimization algorithm and Sobol sequence of points. An interesting benefit using this method is that the same inputs will give the same results. Most of the time this does not happen in meta-heuristic algorithms using random factors. Now the method is modified to find solutions of multimodal optimization problems. Generally in an optimization problem, the differential form of the objective function is needed. In this paper, the proposed method is to find optimum points of general multimodal functions that its differential form is not required. Several problems with benchmark functions have been examined using our method and they give good result

Optimasi Fungsi Multimodal Menggunakan Flower Pollination Algorithm Dengan Teknik Clustering

Optimasi fungsi multimodal merupakan permasalahan yang banyak dijumpai dalam bidang teknik, sains, ilmu sosial dan ekonomi. Tujuan utama dari permasalahan multimodal adalah untuk melokalisir semua solusi yang tersedia baik optimum lokal maupun optimum global dalam sekali running. Flower Pollination Algorithm yang umum digunakan untuk optimasi global perlu dimodifikasi dan dikembangkan agar dapat menyelesaiakan tantangan dalam optimasi fungsi multimodal. Pada penelitian ini kami mengkombinasikan Flower Pollination Algorithm dengan teknik Clustering untuk mengoptimasi fungsi multimodal. Dalam uji coba terhadap 5 fungsi bencharmk multimodal yaitu Second minima, Six hump camel back, Rastrigin, Vincent dan Shubert diperoleh hasil bahwa metode yang disusulkan (FPAC) sukses menemukan semua solusi dari masing-masing fungsi multimodal dalam sekali running baik untuk kasus dimensi rendah maupun dimensi tinggi

An Approximate Optimization Method for Solving Stiff Ordinary Differential Equations With Combinational Mutation Strategy of Differential Evolution Algorithm

This paper examines the implementation of simple combination mutation of differential evolution algorithm for solving stiff ordinary differential equations. We use the weighted residual method with a series expansion to approximate the solutions of stiff ordinary differential equations. We solve the problems from an ordinary stiff differential equation for linear and nonlinear problems. Then, we also implement our method for solving stiff systems of ordinary differential equations. We find that our algorithm can approximate the exact solution of a stiff ordinary differential equation with the smallest error for each length of series that we have chosen. Thus, this approximation method, by using the optimization method of simple combination differential evolution, can be a good tool for solving stiff ordinary differential equations

A Singular Perturbation Problem for Steady State Conversion of Methane Oxidation in a Reverse Flow Reactor

The governing equations describing methane oxidation in a reverse flow reactor are given by a set of convective-diffusion equations with a nonlinear reaction term, where temperature and methane conversion are dependent variables. In this study, the process is assumed to be a one-dimensional pseudohomogeneous model and takes place with a certain reaction rate in which thewhole process ofthereactor is still workable. Thus, the reaction rate can proceed at a fixed temperature. Under these conditions, we can restrict ourselves to solving the equations for the conversion only. From the available data, it turns out that the ratio of the diffusion term to the reaction term is small. Hence, this ratio is considered as a small parameter in our model and this leads to a singular perturbation problem. Numerical difficulties will be found in the vicinity of a small parameter in front of a higher order term. Here, we present an analytical solutionby means of matched asymptotic expansions. The result shows that, up to and including the first order of approximation, the solution is in agreement with the exact and numerical solutions of the boundary value problem

Determination of Gas Pressure Distribution in a Pipeline Network using the Broyden Method

A potential problem in natural gas pipeline networks is bottlenecks occurring in the flow system due to unexpected high pressure at the pipeline network junctions resulting in inaccurate quantity and quality (pressure) at the end user outlets. The gas operator should be able to measure the pressure distribution in its network so the consumers can expect adequate gas quality and quantity obtained at their outlets. In this paper, a new approach to determine the gas pressure distribution in a pipeline network is proposed. A practical and user-friendly software application was developed. The network was modeled as a collection of node pressures and edge flows. The steady state gas flow equations Panhandle A, Panhandle B and Weymouth to represent flow in pipes of different sizes and a valve and regulator equation were considered. The obtained system consists of a set of nonlinear equations of node pressures and edge flowrates. Application in a network in the field involving a large number of outlets will result in a large system of nonlinear equations to be solved. In this study, the Broyden method was used for solving the system of equations. It showed satisfactory performance when implemented with field data