Approximation of the Bessel function of the first kind

Abstract

ذُمذو طشيمح يرسهسهح انمىي، انًشرمح يثاششج يٍ انرعشيف انرحهيهي نذانح تيسم، َرائج دليمح نهغايح في انًُاطك يىثىل نهًماسَح. يع رنك، انمشيثح يٍ َمطح األصم. ذماستها يفهىو جي ذا وسهم انرطثيك، يًا يجعهها يعيا سا ا ذُخفض كفاءذها يع اصدياد ليًح انىسيظ، حيث يرطهة األيش عذ دا أكثش يٍ انحذود نهحفاظ عهً انذلح. يٍ جهح أخشي، ذُمذو طشيمح يىجاخ هاس تذي ل لىي ا يعرًذ عهً دوال أساسيح ثاترح جضئي ا. أظهش هزا انًُهج لذسج عانيح عهً انركيف وكفاءج حساتيح عانيح، ال سيًا نرمشية انذوال عهً فرشاخ أوسع. عهً انشغى يٍ تساطرها، .حممد طشيمح يىجاخ هاس دلح يشضيح تركهفح حساتيح يُخفضح َسثي ا، يًا يجعهها جزاتح نهرطثيماخ انعذديح ذُظهش انًماسَح تيٍ انطشيمريٍ أٌ نكم يُهًا يضاياها انخاصح حسة انسياق. تيًُا ذُف ّضم يرسهسهح انمىي نهرمشيثاخ ذُ أَسة نهرمشيثاخ انشايهح وانًسائم انري ذرطهة حسات ا سشي عا انًحهيح وانذليمح نهغايح، تيًُا طشيمح يىجاخ هاس عّذ تشكم عاو، ذؤكذ هزِ انذساسح أٌ انجًع تيٍ اIn this work, we investigated the approximation of the Bessel function of the first kind using two distinct approaches: the classical power series expansion and the Haar wavelet method. The study highlighted both the theoretical foundations and the practical performance of each technique. The power series method, derived directly from the analytical definition of the Bessel function, provides highly accurate results in regions close to the origin. Its convergence is well understood and straightforward to implement, making it a reliable benchmark for comparison. However, its efficiency decreases for larger values of the argument, where more terms are required to maintain accuracy. On the other hand, the Haar wavelet method offers a powerful alternative based on piecewise constant basis functions. This approach demonstrated strong adaptability and computational efficiency, particularly for approximating functions over larger intervals. Despite its simplicity, the Haar wavelet method achieved satisfactory accuracy with relatively low computational cost, making it attractive for numerical applications. The comparison between the two methods shows that each has its own advantages depending on the context. While the power series is preferable for local and highly precise approximations, the Haar wavelet method is better suited for global approximations and problems requiring fast computation. Overall, this study confirms that combining analytical and numerical techniques provides a deeper understanding of special functions such as the Bessel function.نرمُياخ انرحهيهيح وانشلًيح يُريح فه ًا أعًك نهذوال انخاصح، يثم دانح