Genel görelilikte kullanılan instantonlar, Yang-Mills denklemlerinin sonlu eylem çözümleri olan Yang-Mills instantonlarına karşılık gelen çözümlerdir. Weierstarss’ın genel "yerel en-küçük yüzeyler" çözümü, genel bir instanton metriği verir. Nutku helikoit metriği de bu genel metriğin helikoit en-küçük yüzeyine karşılık gelen özel bir durumudur. Dirac ve Laplace denklemleri dört boyutlu durumda Mathieu fonksiyonları cinsinden çözülebilir. Bir zaman koordinatı metriğe doğrudan eklenirse çözümler, literatürde yüksek boyutlu çözümlerde karşılaşılan çift konfluent Heun fonksiyonları olurlar. Bir dönüşüm yardımıyla Mathieu denkleminin tekillik yapısı elde edilir. Beş boyutlu durum, bu dönüşüm sayesinde Mathieu fonksiyonları cinsinden ifade edilebilir. Zaman koordinatından gelen ek terimle birlikte, radyal ve açısal kısımlar değişik sabitler içerdiğinden, dört boyutlu durumdaki gibi bir ilerletici yazmak için bu fonksiyonların toplanması oldukça zorlaşır. Metriğin orijinde bir eğrilik tekilliğine sahip olması bu bölgenin dışarılanmasını gerektirir. Bu da uygun sınır koşullarının kullanımını önemli kılar. Tek sayılı boyutlarda, Atiyah, Patodi ve Singer tarafından tanımlanmış olan yerel olmayan spektral sınır koşulları, topolojik engeller sebebiyle zorunludur. Çift sayılı boyutlarda yerel sınır koşulları kullanılabilse de, Dirac operatörünün ve yük eşleniği simetrilerinin korunması isteniyorsa yerel olmayan spektral sınır koşulları kullanılmalıdır. Bu problemde sınır koşulları uygulanırken Atiyah-Patodi-Singer formalizmi kullanılmıştır. Manifoldun sınırında yazılan denklemler, sınır tanımlanmadan yazılan denklemlerden daha tekildir ve bu da çözümü zorlaştırır. Bilgisayar, denklemlerin çıkarılması ve analizlerinde yoğun olarak kullanılmıştır. Newman-Penrose formalizmini kullanan bir Maple paketi, çalışmadaki analitik hesapları yapmak için geliştirilmiştir. Paket ayrıca instanton metrikleri için tam bir Newman-Penrose hesaplayıcısı olarak kullanılabilir. Anahtar Kelimeler: Dalga denklemleri, Atiyah-Patodi-Singer sınır koşulları, Heun fonksiyonları, sembolik hesaplama.The interest in higher dimensional wave equations is driven by the usage of higher dimensional metrics in general relativity and string theory. Instanton solutions of general relativity are the counterparts of Yang-Mills instantons which are finite-action solutions of the Yang-Mills equations. They have an important contribution to the path-integral in the quantization of the Yang-Mills fields. The general relativistic instantons are also expected to play a similar role in the path-integral approach to quantum gravity. Weierstrass' general local solution of minimal surfaces yields to a general instanton metric and Nutku's helicoid metric is a special case which corresponds to the helicoid minimal surface of this general metric. Dirac and Laplace equations can be solved in terms of Mathieu functions in the four dimensional case. If a time coordinate is added trivially to the metric, the solutions become double confluent Heun functions which are known to arise in higher dimensional solutions in the literature. One can trade the irregular singularity at zero by two regular singularities at plus and minus one by a transformation, to reach at the same singularity structure of the Mathieu equation and give the solutions in this form. But the main difference between the two cases is that, although both the radial and the angular parts can be written in terms of Mathieu functions, the constants are different, modified by the presence of the new term coming from the time-dependence, which makes the summation of these functions to form the propagator quite difficult. In four dimensions one can use the summation formula for the product of four Mathieu functions -two of them for the angular and the other two for the radial part- summing them to give us a Bessel type expression. Nutku helicoid solution has a curvature singularity at the origin. Therefore, in order to have a precise result we tried to apply the Atiyah-Patodi-Singer spectral boundary conditions which was necessitated by the dimension of the spacetime. One is free to choose the local boundary conditions in even dimensions. However, we applied the same type of boundary conditions to conserve and charge conjugation symmetries of the Dirac operator in the four dimensional case. The application of the spectral boundary conditions involves the solution of the so called the little Dirac equation, the Dirac equation written on the boundary of the manifold. The analytical solutions of the little Dirac equation could not be obtained. The singularity structure of the little Dirac equation can reveal why we could not obtain an analytical solution. The singularity analysis shows that the equation has irregular singularities at zero and infinity and a regular singularity at minus one. We see that this equation has one more singularity than the Heun functions; thus, our solution is not one of the better known solutions in the literature, which are included in the computer packages like Maple or cited in the comprehensive books such as Ince's. The computer programs are involved intensively in the derivation and analysis of the equations. Newman-Penrose (NP) formalism is a strong technique for investigating the physical properties of the exact solutions of Einstein's field equations. Goldblatt has developed NP formalism for gravitational instantons. In the gravitational instanton case, the gravitational field decomposes into its self-dual and anti-self-dual parts and this decomposition is natural in the spinor approach which necessitates two independent spin frames for the spinor structure of 4-dimensional Riemann manifolds with Euclidean signature. A Maple package using the Goldblatt's Newman-Penrose formalism for instanton spaces is developed for the analytical computations needed in the work. The package also supplies a complete Newman-Penrose calculator for instanton metrics. Although the numerical methods are still important for the analytically non-solvable systems such as N-body dynamics, analytical methods are getting more and more crucial in the applicable cases as the capacity of the computer systems and the symbolic manipulation packages increase. It is a strong estimate that we will encounter higher equations -in the sense of the singularity structure- than the hypergeometric equations gradually more in the literature, as the phenomena in higher dimensions or in different geometries are studied. Having different confluent types and special cases which reduce to less singular equations, Heun's equation has a special importance to bring together the known literature with the physics of the future. Keywords: Wave equations, Atiyah-Patodi-Singer boundary conditions, Heun functions, symbolic computation
