A Finite Difference Method for Numerical Solution of Goursat Problem of Partial Differential Equation

The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes several finite difference schemes for solving the one-dimensional convection-diffusion equation with constant coefficients. In this research the use of modified equivalent partial differential equation (MEPDE) as a means of estimating the order of accuracy of a given finite difference technique is emphasized. This approach can unify the deduction of arbitrary techniques for the numerical solution of convection-diffusion equation. It is also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one-dimensional constant coefficient convection-diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

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Analisis Lanjut Metode Beda Hingga Eksplisit Untuk Menentukan Harga Opsi

Unisda Journal of Mathematics and Computer Science (UJMC) ◽

10.52166/ujmc.v5i01.1302 ◽

2019 ◽

Vol 5 (01) ◽

pp. 41-46

Author(s):

Wahyudi Sastro

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Finite Difference ◽

Finite Difference Method ◽

Option Pricing ◽

Difference Method ◽

A Value ◽

Black Scholes ◽

Explicit Finite Difference ◽

Explicit Finite Difference Method

Abstract. Explicit finite difference method is used to approximate a partial differential equation that is applied to determine the option pricing. The results of this study note that the calculation of option pricing using explicit finite difference method is negative when partition N ≥ 25 with a value of -2.21. Thus, the results of the calculation of option pricing are not convergent and away from the results of analyzing the option pricirng (Black-Scholes) solution. This is because one of the three probabilities Bj = 1- σ2j2Δt is negative, namely (-0.12) when j ≥ 12 with S ≥ 16.25 (in units). So this explicit finite difference method cannot be used to determine the option pricing. Keywords: Option Pricing, Explicit Finite Difference Method Abstrak. Metode beda hingga eksplisit digunakan untuk mengaproksimasi suatu persamaan diferensial pasial yang aplikasikan untuk menentukan harga opsi. Hasil penelitian ini diketahui bahwa perhitungan harga opsi dengan menggunakan metode beda hingga eksplisit bernilai negatif pada saat partisi N ≥ 25 dengan nilai -2,21. Dengan demikian, hasil perhitungan harga opsi tidak konvergen dan menjauhi hasil solusi analitik perhitungan harga opsi (Black-Scholes). Hal ini disebabkan karena salah satu ketiga probabilitas Bj = 1- σ2j2Δt yaitu bernilai negatif yaitu (-0.12) saat j ≥ 12 dengan S ≥ 16.25 (dalam satuan). Sehingga metode beda hingga eksplisit ini tidak dapat digunakan untuk menentukan harga opsi. Kata Kunci: Harga Opsi, Metode Beda Hingga Eksplisit.

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